Almost bi-Lipschitz embeddings and almost homogeneous sets

TRANSACTIONS OF THE AMERICAN MA THEMA TICAL SOCIETY V olume 00, N umber 0, Pages 000– 000 S 0002-9947(XX)0000 -0 ALMOST BI-LIPSCHITZ EMBEDDINGS AND ALMOST HOMOGENE OUS SETS ERIC J. OLSON AND JAM ES C. ROBINSON Abstract. This pap er is concerned with em beddings of homogeneous spaces int o Eu clidean spaces. W e sho w tha t any homoge neous metric space can be e m- bedded into a Hilbert spa ce using an almost bi-Lipsc hitz mapping ( bi-Lipschitz to within logarithmic corrections). The image of this set is no longer homo- geneous, but ‘almost homogeneous’. W e therefore study the problem of em- bedding an almost homogeneou s subset X of a Hilbert space H i n to a finite- dimensional Euclidean space. In fact w e show that i f X is a compact subset of a Banac h space and X − X is almost homogeneous then, for N sufficien tly large, a pr ev alent set of li near m aps fr om X int o R N are al most bi- Lipsc hitz betw een X and its i mage. W e ar e then able to use the K urato wski embedding of ( X, d ) i n to L ∞ ( X ) to prov e a simi lar result for compact metric spaces. 1. Introduction In this pa per w e in v estigate abstr act embeddings b etw een metric spaces , Hilb ert spaces, a nd finite-dimensio nal E uclidean spa ces. Histor ically (star ting with Bouli- gand in 1 928), a tten tion has b een o n bi-Lips ch itz embeddings . By weakening this to almost bi-Lipschitz em beddings we ar e able to obtain a num ber o f new results. A metric space ( X, d ) is said to be ( M , s )- homo gene ous (or simply homo gene ous ) if a n y ball of r adius r ca n b e cov er ed by at most M ( r/ρ ) s smaller balls o f r adius ρ . Since any subset o f R N is homogeneo us and homog eneit y is preserved under bi- Lipschitz mapping s, it follows tha t ( X , d ) must b e homogeneous if it is to admit a bi-Lips ch itz embedding in to some R N (cf. comment s in Ha j lasz, 2 003). The Assouad dimension of X , d A ( X ), is the infimum o f a ll s such that ( X , d ) is ( M , s )- homogeneous for some M ≥ 1. Assouad (1 983) showed that ( X , d ) is homogeneous if a nd only if the snowflak e spaces ( X , d α ) with 0 < α < 1 admit bi-Lips c hitz em bedding s int o some R N (where N dep ends on α ). Howev e r, there are exa mples due to Laa kso (200 2; see also Lang & Pla ut, 2 001) of homo geneous spac es that do not admit a bi-Lipschitz em bedding int o any R N , nor even into an infinite-dimensio nal Hilb ert s pace. This pa per starts with a simple result, based on Asso uad’s arg ument , that any homogeneous metric space admits an almost bi-Lipschi tz em bedding in to an infinite-dimensiona l Hilb ert space. Receiv ed b y the editors March 2005. 1991 M athematics Subject Classific ation. 54F45 , 57N35. Key wor ds and phr ases. Assouad dimension, Bouligand dimension, Doubling spaces, Embed- ding theorems, Homogeneous s paces. JCR is a Roy al So ciety University Researc h F ellow, and would l ike to thank the Society for al l their supp ort. c  1997 American Mathematical So ciety 1 2 ERIC J. OLSON AND JAMES C. ROBINSON The cla ss of γ - almost L -bi-Lipschitz mappings f : ( X , d ) → ( ˜ X , ˜ d ) (or almost bi-Lipschitz mappings for short) consists of all those maps for which ther e exists a γ ≥ 0 and an L > 0 such that (1.1) 1 L d ( x, y ) slog( d ( x, y )) γ ≤ ˜ d ( f ( x ) , f ( y )) ≤ L d ( x, y ) for all x, y ∈ X suc h that x 6 = y . Here slog( x ) is the ‘symmetric loga rithm’ of x , defined as slog( x ) := lo g( x + x − 1 ) , and so an a lmost bi-Lipschitz map is bi-Lipschitz to within lo garithmic correc tions. Although the bi-L ipschit z image of a homog eneous s et is homogene ous, this is not tr ue for almost bi-Lipsch itz imag es; they a re, how ev er, almost homo gene ous : we say that ( X, d ) is ( α, β )-almost ( M , s )-homogeneous if (1.2) N X ( r , ρ ) ≤ M  r ρ  s slog( r ) β slog( ρ ) α for a ll 0 < ρ < r < ∞ . The Assouad ( α, β ) -dimension o f X , d α,β A ( X ), is the infim um of all s such that X is ( α, β )-almost ( M , s )-homogeneous for some M ≥ 1. Olson (2002 ) show ed that given a compact X ⊂ R N with d A ( X − X ) = d then almost every pro jection of rank k > d provides a n almost bi-Lips c hitz embedding of X into R k . In this pa per w e s how a simila r result for compact subsets X of a Hilb ert spa ce: If the set of differences 1 X − X is a lmost homogeneous with d α,β A ( X − X ) = d then ‘most’ linea r maps into E uclidean spaces R k of with k > d provide a lmost bi-Lipschitz e m beddings of X . More explicitly , if k > d then the set of almos t bi-L ipschit z em beddings int o R k is prev alent in the space o f a ll linea r maps into R k , in the sense of Hun t, Sauer & Y orke (1992 ). W e then extend this result to subsets o f Banach spa ces. There is an unfor tunate gap here. An a lmost homogeneo us metric space has an almost bi-Lipschitz image that is a n almost homoge neous subset of a Hilb ert spa ce. How ever, our embedding theorem for a subset X of a Hilb ert space requires tha t not X itse lf, but the s et X − X of differences is almo st homogeneous. By using the K uratowski isometric em bedding of ( X , d ) into L ∞ ( X ) we ca n as- sign a meaning to “ X − X ” even when X is a metric spac e. With this interpretation, we can also s how that if ( X, d ) is a co mpact metric spa ce then the assumption that X − X is almo st homogeneous is sufficient to ensure that ( X , d ) ca n b e embedded int o a Euclidean space in an almost bi-Lipschitz w a y . In Section 2 w e state some elementary prop erties of the ( α, β )-Assouad dimension and sho w tha t an y almo st ho mogeneous metric space ( X , d ) can be embedded into a Hilb ert spac e in an almo st bi-Lipschitz w ay; that such almost bi-Lipschitz images of almost ho mogeneous spaces are aga in almost homo geneous is shown in Section 3. Section 4 treats the loca l versions of homogeneity and almost ho mogeneity . Section 5 con tains our m ain result on em b edding a subset X of a Hilbert space with 1 The i n troduction of a condition on the dimension of the set X − X of differences, r ather than on X itself , i s common in the l iterature on abstract embeddings. The pro of of M a˜ n ´ e’s 1981 em bedding theorem requires t he Hausdorff dimension of X − X to b e finite , a condition no t ensured b y the finitene ss of d H ( X ). F oias & Olson (1996) and Hun t & Kaloshin (1999) treat the upper box-coun ting dimension whi c h is unusual in having the prop erty that d F ( X ) < ∞ impl ies that d F ( X − X ) < ∞ . [Recall that d F ( X ) = l im sup ǫ → 0 log N ( X , ǫ ) / ( − log ǫ ), where N ( X, ǫ ) is the minimum num ber of balls of radius ǫ needed to co v er X .] ALMOST BI-LIPSCHITZ EMBEDDINGS 3 X − X almost homogeneous, while in Sectio n 6 we consider what is p ossible for suc h subsets knowing only pro per ties of X . In Sectio n 7 we generalise o ur main theor em to trea t s ubsets of Banach spaces , and hence g ive a result for compact metric spaces. In Section 8 we explore the r elationship betw een d α,β A ( X ) and d α,β A ( X − X ). After Section 9 , wher e we give an exa mple of a ho mogeneous set that cannot b e bi-Lipschitz em bedded into an y R k using a n y line ar map, w e finish with some int eresting op en problems. 2. Almost homogeneous metric sp aces As discussed ab ove, we will s ay that a metr ic space ( X , d ) is ( α, β )-almos t ( M , s )- homogeneous (or simply almost homogeneous) if an y ball of radius r can be covered by at most 2 (2.1) N X ( r , ρ ) ≤ M  r ρ  s slog( r ) β slog( ρ ) α balls of radius ρ (with ρ < r ), for some M ≥ 1 and s ≥ 0, where slog( x ) = log( x + x − 1 ). W e now give so me simple prop erties of the function slog. Lemma 2. 1. Given L > 0 and γ ≥ 0 , t her e exist c onstants A L , B L , a γ , b γ , σ ∈ (0 , ∞ ) indep endent of x su ch t hat (p1) | log( x ) | ≤ slog( x ) ≤ log 2 + | log ( x ) | , in p articular slo g(2 k ) ≤ (1 + | k | ) log 2 , (p2) A L slog( x ) ≤ slog ( Lx ) ≤ B L slog( x ) , (p3) a γ slog( x ) ≤ slog( x slo g( x ) γ ) ≤ b γ slog( x ) , for al l x ≥ 0 , and (p4) if 2 − ( k +1) ≤ x ≤ 2 − k then slog( x ) ≥ σ slog(2 − k ) . Pr o of. (p1) is elemen tary . F or (p2) consider the quotien t function g : (0 , ∞ ) → (0 , ∞ ) defined by g ( x ) = slog( Lx ) slog( x ) . Let a L = inf { g ( x ) : x ∈ (0 , ∞ ) } and b L = sup { g ( x ) : x ∈ (0 , ∞ ) } . Since lim x → 0 g ( x ) = 1 , lim x →∞ g ( x ) = 1 , and 0 < g ( x ) < ∞ for x ∈ (0 , ∞ ) , then b oth a L and b L are finite p ositive constant s. The proo f o f (p3) is similar. F or (p4) set x = 2 − r with k ≤ r ≤ k + 1. Since slog ( x ) = log( x + 1 /x ) ≥ log 2 and slog(2 − r ) ≥ | log 2 − r | = | r | log 2 from (p1), then slog ( x ) ≥ (1 + | r | ) / 2. Therefor e, the estimate slog(2 − k ) slog( x ) ≤ (1 + | k | ) log 2 (1 + | r | ) / 2 ≤ 4 log 2 gives (p4) with σ = 1 / (4 log 2 ).  2 F or bounded metric spaces (2.1) could be replaced by N X ( r , ρ ) ≤ M ′ „ r ρ « s log(e + ρ − 1 ) γ , (in terms of our curr en t definition we w ould ha v e M ′ ≥ M and γ = α + β ) whi le for compact spaces the f actor of e in the logarithm could also b e dropped by considering only ρ ≤ r ≤ ǫ for some ǫ > 0 (see Section 4). Ho w ev er, (2.1) all o ws us to treat general metric spaces. 4 ERIC J. OLSON AND JAMES C. ROBINSON W e define the Asso uad ( α, β )-dimension of X , d α,β A ( X ), to b e the infim um of all s for which X is ( α, β )-almost ( M , s )-homog eneous. When α = β = 0 we recov er the standard definition of a homogeneo us space a nd the usual Assouad dimension. W e note her e that it is s traightforw ard to s how tha t the Assouad ( α, β )-dimension satisfies the minimal pro per ties we would ask for in a dimension, namely that X ⊆ Y ⇒ d α,β A ( X ) ≤ d α,β A ( Y ) , d α,β A ( X ∪ Y ) = max( d α,β A ( X ) , d α,β A ( Y )) , and d α,β A ( O ) = n if O is an op en subset of R n . F urthermore, (2.2) α 1 ≥ α 2 and β 1 ≥ β 2 ⇒ d α 1 ,β 1 A ( X ) ≤ d α 2 ,β 2 A ( X ) . W e now show t hat if ( X , d ) is almos t homogeneous then it ca n be embedded into an infinite-dimensional Hilber t space in an almost bi-Lipschitz wa y . Key to this result is the following prop osition, which althoug h not given explicitly in this form, essentially o ccurs in Assouad’s pap er . Indeed, it is the main ingredient in his pro of of the existence of bi-L ipschit z maps b etw e en ( X, d α ) and R N . Prop ositio n 2 .2. L et ( X, d ) b e an ( α, β ) -almost ( M , s ) - homo gene ous metric sp ac e and distinguish a p oint a ∈ X . Th en ther e ar e c onstants A, B , C > 0 such that for every j ∈ Z t her e exists a map φ j : ( X, d ) → R M j , wher e M j = C (1 + | j | ) α + β , with φ j ( a ) = 0 , and for every x 1 , x 2 ∈ X (a1) 2 − ( j +1) < d ( x 1 , x 2 ) ≤ 2 − j implies that k φ j ( x 1 ) − φ j ( x 2 ) k ≥ A , and (a2) k φ j ( x 1 ) − φ j ( x 2 ) k ≤ B M j min[1 , 2 j d ( x 1 , x 2 )] . Pr o of. The pro of follows exa ctly the s teps in Assouad’s or iginal pap er (see also the lecture notes of Heinonen (20 03) for an account that is ea sier to fo llow) which we outline very briefly here: if N j is a ma ximal 2 − j net in ( X , d ), then for every x ∈ X card  N j ∩ B ( x, 1 2 · 2 − j )  ≤ N X (12 · 2 − j , 2 − j − 1 ) ≤ 24 M slo g(12 · 2 − j ) α slog(2 − j − 1 ) β ≤ C (1 + | j | ) α + β where the constant C is a pro duct of M and the co nstants app earing in Prop o- sition 2.1. Thus, there exists a ‘colouring map’ κ j : N j →  e 1 , . . . , e M j  , where  e 1 , . . . , e M j  is the standard basis of R M j , s uc h that κ j ( a ) 6 = κ j ( b ) if d ( a, b ) < 12 · 2 − j . Le t ˜ φ j ( x ) = X a i ∈ N j max  (2 − 2 j d ( x, a i )) , 0  κ j ( a i ) . Note that 2 2 − j < d ( x 1 , x 2 ) ≤ 2 3 − j implies ˜ φ j ( x 1 ) is orthogona l to ˜ φ j ( x 2 ). It is then straightforw ard to show that the map φ j ( x ) = ˜ φ j +3 ( x ) − ˜ φ j +3 ( a ) satisfies the prop erties given in the statement of the prop osition.  Theorem 2.3. L et ( X , d ) b e an ( α, β ) - almost ( M , s ) -homo gene ous m etric sp ac e and H an infinite-dimensional s ep ar able Hilb ert sp ac e. Then, for every γ > α + β + 1 2 , ther e exists a map f : X → H and a c onst ant L such that 1 L d ( x, y ) slog( d ( x, y )) γ ≤ k f ( s ) − f ( t ) k ≤ L d ( x, y ) , i.e., f is γ -almost bi-Lipschitz. ALMOST BI-LIPSCHITZ EMBEDDINGS 5 Pr o of. Le t { e j } j ∈ Z be a n o rthonormal set of vectors in some Hilbert space. Let δ > 1 / 2 and define f : ( X , d ) → L ∞ j =1 R M j ⊗ e j ≃ H by (2.3) f ( x ) = ∞ X j = −∞ 2 − j (1 + | j | ) δ M j φ j ( x ) ⊗ e j , where the maps φ j are those of Pr op osition 2.2. Since f ( a ) = 0, then the uppe r bo und on k f ( s ) − f ( t ) k that we now prov e will also show convergence o f the se- ries (2.3) defining f . Let ( x 1 , x 2 ) b e a pair of distinct p oints o f X . Thus, there exists l ∈ Z such that 2 − ( l +1) < d ( x 1 , x 2 ) ≤ 2 − l . Note tha t for such a pair of points k φ l ( x 1 ) − φ l ( x 2 ) k ≥ A . W e hav e k f ( x 1 ) − f ( x 2 ) k 2 = ∞ X j = −∞ 2 − 2 j (1 + | j | ) 2 δ k φ j ( x 1 ) − φ j ( x 2 ) k 2 M 2 j ≤ ∞ X j = −∞ B 2 (1 + | j | ) 2 δ d ( x 1 , x 2 ) 2 ≤ c 1 d ( x 1 , x 2 ) 2 , where the sum conv erges since 2 δ > 1 . The low er bound is str aightforw ard, since k f ( x 1 ) − f ( x 2 ) k ≥ 2 − l (1 + | l | ) δ M l k φ l ( x 1 ) − φ l ( x 2 ) k ≥ A 2 − l (1 + | l | ) δ M l ≥ c 2 2 − l (1 + | l | ) α + β + δ ≥ c 2 d ( x, y ) (1 + | l | ) α + β + δ Since d ( x, y ) = 2 − r with l ≤ r < l + 1 it follows us ing (p1) from Lemma 2 .1 that 1 + | l | slog( d ( x, y )) = 1 + | l | slog(2 − r ) ≥ 1 + | l | (1 + | r | ) log 2 ≥ 1 2 log 2 , and so k f ( x 1 ) − f ( x 2 ) k ≥ c 3 d ( x, y ) slog( d ( x, y )) α + β + δ . T aking L = max( c 1 , 1 /c 3 ) finishes the pro o f.  W e note here that if ( X , d ) is bounded then there exists a k such that d ( x 1 , x 2 ) ≤ 2 k for all x 1 , x 2 ∈ X . In this case the definition of f in (2.3) can b e s implified to (2.4) f ( x ) = ∞ X j = − k 2 − j (1 + | j | ) δ M j φ j ( x ) ⊗ e j and will still provide a γ -almost bi-Lipschitz em bedding. 3. Almost bi-Lipschitz images of sets Since we can embed a n y almost homogeneo us metric space into a Hilb ert space using an a lmost bi-Lips c hitz map, it is natural to study the effect of such mappings on a lmost homoge neous spaces . Here we show tha t a lmost bi-Lipschitz images of almost homog eneous metric space s ar e still a lmost homo geneous. In particula r this implies th at it is necessa ry tha t X be almost homogeneous if it is to enjoy an almost bi-Lipschitz embedding into so me R N . 6 ERIC J. OLSON AND JAMES C. ROBINSON Lemma 3.1. L et ( X , d ) b e an ( α, β ) -almost ( M , s ) -homo gene ous m et ric sp ac e and φ : ( X , d ) → ( ˜ X , ˜ d ) a γ -almost L -bi-Lipschitz map. Then ( φ ( X ) , ˜ d ) is an almost homo gene ous metric sp ac e with d α + γ ,β + γ A ( X ) ≤ d α,β + γ A ( φ ( X )) ≤ d α,β A ( X ) . Pr o of. Incr ease L if necessar y so that (3.1) L 2 b γ (log 2) γ ≥ 1 , where here and in the r est of the pro of b = b γ , whe re b γ is the consta n t occurr ing in (p3) in Lemma 2.1; clearly φ rema ins γ -almost L -bi-Lipschitz under this assump- tion. T ake s > d α,β A ( X ), 0 < ρ < r < ∞ , a nd consider an arbitrary ball B ˜ X ( φ ( x ) , r ) of radius r in φ ( X ). Now, we ha ve B ˜ X ( φ ( x ) , r ) ⊆ φ { B X ( x, Lr b γ slog( Lr b γ ) γ ) } , since using (p3) in Lemma 2 .1 1 L Lr b γ slog( Lr b γ ) γ slog( Lr b γ slog( Lr b γ ) γ ) γ ≥ rb γ slog( Lr b γ ) γ [ b slog ( Lrb γ )] γ = r . By o ur choice o f L in (3.1) and since ρ < r we have 0 < ρ/L < Lr b γ slog( Lr b γ ) γ and so we ca n cov e r B X ( x, Lr b γ slog( Lr b γ ) γ ) by N X ( Lrb γ slog( Lr b γ ) γ , ρ/L ) ≤ M  Lr b γ slog( Lr b γ ) γ ρ/L  s slog( Lr b γ slog( Lr b γ ) γ ) β slog( ρ/L ) α ≤ c 1  r ρ  s slog( r ) β slog( ρ ) α balls of r adius ρ/L (in X ) where c 1 depe nds on M , L and the constants app ear ing in Lemma 2.1. Denote these balls by B X ( x i , ρ/L ). Since φ { B X ( x i , ρ/L ) } ⊆ B ˜ X ( φ ( x i ) , ρ ) and B ˜ X ( φ ( x ) , r ) was arbitrary , it follows that N φ ( X ) ( r , ρ ) ≤ c 1  r ρ  s slog( r ) β slog( ρ ) α for any 0 < ρ < r < ∞ . Thu s φ ( X ) is ( α, β + γ )-almost ( c 1 , s )-homog eneous. T aking the infimum ov er s > d α,β A ( X ) yields d α,β + γ A ( φ ( X )) ≤ d α,β A ( X ). By co nsidering s imilarly the inv erse map φ − 1 : φ ( X ) → X one obtains the low er bo und d α,β + γ A ( φ ( X )) ≥ d α + γ ,β + γ A ( X ).  Combined with Lemma 3.1, the embedding res ult o f Prop osition 2.3 s hows that any almos t homog eneous metric space ( X , d ) has a n almos t bi-Lipschitz image f ( X ) that is an almost homo geneous subset of a Hilb ert space. W e end by noting sinc e a lmost bi-Lipsc hitz ma ps are, in fact, Lipschitz then for any almos t bi-Lipschitz ma p φ the upper box-counting (‘fractal’) dimension sa tisfies d F ( φ ( X )) ≤ d F ( X ). Moreover, it is not difficult to prove the following: Lemma 3.2. L et ( X , d ) b e a metric sp ac e and φ : ( X, d ) → ( ˜ X , ˜ d ) an almost bi-Lipschitz map. Then d F ( φ ( X )) = d F ( X ) . ALMOST BI-LIPSCHITZ EMBEDDINGS 7 4. Aside: C omp act sp aces and local versions of (almost) homogeneity In this section we briefly discuss the lo cal definitions o f homog eneity and al- most homo geneity , a nd the dimensions asso ciated with them. While they agree for compact spaces, they ar e distinct in gener al. A metr ic s pace ( X , d ) is sa id to be lo c al ly ( M , s ) -homo gene ous (o r simply lo c al ly homo gene ous ) if ther e ex ists an ǫ > 0 such that a ny ball of r adius r < ǫ can b y cov ered b y a t mos t M ( r /ρ ) s smaller balls of radius ρ . The in troduction of the constant ǫ for a lo cally ho mogeneous s pace may b e interpreted as the small s cale bene ath which the s et may b e v iewed as homoge neous. In this ca se M may dep end on ǫ which in turn dep ends on the units of measur emen t used in the definition of the metric. Mov a hedi-Lank ar ani (19 92) defined the metric (or ‘Bouligand’) dimension (4.1) d B ( X ) = lim ǫ → 0 lim t →∞ sup  log N X ( r , ρ ) log( r /ρ ) : 0 < ρ < r < ǫ and r > tρ  , where N X ( r , ρ ) is the minimum n um ber of balls of radius ρ necess ary to co v er a n y ball of radius r . This dimension, d B ( X ), is the infimum of all s such that ( X, d ) is lo cally ( M , s )-homog eneous for some M ≥ 1 . Here we give a simple example that sho ws that the concepts of homogeneous and loc ally homo geneous are indeed differe n t. Let H be a Hilber t spa ce with orthonor mal basis g iven by { e n } n ∈ N . Define X = { ρ n e n : n ∈ N } where ρ n = 1 − 1 n . If ( X , d ) is ( M , s )-homo geneous for some M and s then (4.2) N X ( ρ 2 n , ρ n ) ≤ M ( ρ 2 n /ρ n ) s = M  2 n − 1 2 n − 2  s ≤ M . How ever, e ach ball B (0 , ρ 2 n ) contains the n p oints { 0 } ∪ { ρ k e k : n < k < 2 n } which ar e mutually more than a distance ρ n apart. Therefor e N X ( ρ 2 n , ρ n ) ≥ n . T aking n large eno ugh shows that (4.2) cannot hold, and s o ( X, d ) is no t homoge- neous. O n the other hand, ( X , d ) is lo cally homogeneo us for an y ǫ < 1. Note that if ( X , d ) is c ompact, then the notions of homogeneous and locally homogeneous are equiv alent (see Olson, 2002). Thus d A ( X ) = d B ( X ) for compact spaces X . As with homoge neous spaces, there is a similarly distinct notion of lo c al ly ( α, β ) - almost ( M , s ) -homo gene ous . This means there is s ome ǫ > 0 such that (2.1) holds for all 0 < ρ < r < ǫ . Similar arg umen ts to those g iven in Olson (20 02) show that the notions o f almost homogeneous and lo cally almost homogeneous ar e equiv- alent when ( X, d ) is c ompact. Define the lo c al Assouad ( α, β ) -dimension o f X , d α,β B ( X ) , to b e the infim um of all s such that ( X , d ) is lo ca lly ( α, β )-almo st ( M , s )- homogeneous for some ǫ > 0 and M ≥ 1. Let ( X , d ) be a metric spa ce. In g eneral d α,β B ( X ) ≤ d α,β A ( X ). Both d α,β A and d α,β B are inv a riant under a r escaling of the metric. Thus, the metric space ( ˜ X , ˜ d ) where ˜ X = X and ˜ d = η d for so me η > 0 ha s d α,β A ( ˜ X ) = d α,β A ( X ) and d α,β B ( ˜ X ) = d α,β B ( X ). Note that d α + θ β , (1 − θ ) β B ( X ) ≤ d α,β B ( X ) ≤ d (1 − θ ) α,θ α + β B ( X ) 8 ERIC J. OLSON AND JAMES C. ROBINSON for 0 ≤ θ ≤ 1. Mo reov er, if X is compact, then d F ( X ) ≤ d α,β A ( X ) = d α,β B ( X ) where d F ( X ) denotes the fra ctal or upper b ox-coun ting dimens ion. W e no te here that d B shares with d A the usual pro pe rties of dimension discusse d in Section 2, along with the monotonicty pr op erty in (2.2). 5. Embedding Hilber t subsets X with X − X homogeneous In this section w e prove our main result, in which we ta ke a subspace X of a Hilber t space, assume that X − X is almost ho mogeneous, and obtain an almost bi-Lipschitz embedding into a finite-dimens ional space. Our argument is es sent ially a com bination of that o f Olson (2002), who treated a subset X o f a Euclidea n space w ith d A ( X − X ) finite, a nd that of Hunt & K aloshin (1999), who considered a s ubset of a Hilb ert space with finite upp er b ox-count ing (‘fractal’) dimension. The k ey to com bining th ese successfully is Lemma 5.3, b elow. In line with the trea tmen t in Sauer et al. (19 91) and in Hun t & Kalo shin (1999), our main theorem is express ed in terms o f pre v alence. This concept, which gen- eralises the notion of ‘almost e very’ from finite to infinite-dimens ional spaces, w as int ro duced b y Hun t, Sauer & Y orke (1992 ); see their pap er for a detailed discussion. Definition 5. 1. A Borel subset S of a no rmed linear space V is pr evalent if there exists a compactly supp orted pr obability meas ure µ such that µ ( S + v ) = 1 for a ll v ∈ V . In particular, if S is prev alen t then S is dense in V . Note that if we set Q = supp( µ ) then Q can be thoug h t of as a ‘probe s et’, which consists of ‘a llow a ble per turbations’ with which , g iven a v ∈ V , we ‘prob e’ and tes t whether v + q ∈ S for a lmost every q ∈ Q . Since we will use it b elow, a nd for its historica l imp ortance, we quote Hunt & Kaloshin’s r esult here, in a form suitable fo r what follo ws. Given a s et X , w e recall here that its upp er b ox-counting (‘fra ctal’) dimension is defined as d F ( X ) = lim sup ǫ → 0 log N ( X , ǫ ) − log ǫ , where N ( X , ǫ ) denotes the minim um num ber of ba lls o f r adius ǫ necessary to cov er X ; and its thickness exponent, τ ( X ), is (5.1) τ ( X ) = lim sup ǫ → 0 log d ( X , ǫ ) − log ǫ , where d ( X , ǫ ) is the minim um dimension of a ll finite-dimensional subspac es, V , of B such that every p oint of X lies within ǫ of V . W e note her e for later use that τ ( X ) ≤ d F ( X ). Theorem 5. 2 (Hunt & Kalo shin) . L et X b e a c omp act subset of a Hilb ert sp ac e H , D an inte ger with D > d F ( X − X ) , and τ ( X ) the thickness exp onent of X . If θ is chosen with θ > D (1 + τ ( X ) / 2 ) D − d F ( X − X ) then for a pr evalent set of line ar maps L : B → R D ther e exist s a c > 0 such that c k x − y k θ ≤ | Lx − L y | ≤ k L k k x − y k for al l x, y ∈ X ; in p articular these maps ar e inje ctive on X . ALMOST BI-LIPSCHITZ EMBEDDINGS 9 W e note here that d F ( X − X ) ≤ 2 d F ( X ), so tha t fo r zer o thickness sets with finite box-counting dimension one can c ho ose an y D > 2 d F ( X ) and θ > D / ( D − 2 d F ( X )). 5.1. Construction of the probabili t y measure µ for a giv en X . W e now apply the definition of prev alence given a particular compa ct subset X of our Hilbert space H suc h that X − X is ( α, β )-almost ( M , s )-homogeneous. F or so me fixed N , let V be the s et of linear functions L : H → R N . W e now construct a compactly suppo rted probability measure µ on V (a s r equired by the definition of pr ev alence) tha t is carefully tailore d to the particular set X . Key to this is the following result. Lemma 5.3. Su pp ose that X is a c omp act ( α, β ) -almost ( M , s ) -homo gene ous s ubset of H . Then ther e ex ists a se quenc e of nest e d line ar subsp ac es U n with U n ⊆ U n +1 , dim U n ≤ C (1 + n ) α + β +1 , and k P n x k ≥ 1 8 k x k for al l x ∈ X with k x k ≥ 2 − n , wher e P n is the ortho gonal pr oje ction onto U n . Pr o of. Co nsider the collection of shells ∆ j = n x ∈ X : 2 − ( j +1) ≤ k x k ≤ 2 − j o . Since ∆ j ⊂ B (0 , 2 − j ) it can be cov ered using N X (2 − j , 2 − ( j +3) ) ≤ 8 s M (log 2) 2 (1 + | j | ) β (4 + | j | ) α ≤ c 2 (1 + | j | ) α + β := M j balls o f r adius 2 − ( j +3) , where c 2 is indep endent o f j . W e choos e the c en tres n u ( j ) i o M j i =1 of these balls so that k u ( j ) i k ≥ 2 − ( j +2) . Since X is compact, X ⊂ B (0 , 2 k ) for some k sufficiently larg e, a nd so n [ j = − k ∆ j =  x ∈ X : k x k ≥ 2 − n  . Let P n be the orthogona l pro jection onto the linear subspac e U n spanned by the collection n u ( j ) i : j = − k , . . . , n and i = 1 , . . . , M j o . Then the dimension of U n is bo unded by c 3 (1 + n ) α + β +1 using the same estimate a s in (6.1). Moreov er, for every x ∈ ∆ j there exists u ( j ) i such that x = u ( j ) i + v where k v k ≤ 2 − ( j +3) . Since k P n k = 1 a nd k P n u k = k u k for u ∈ U n , then k P n x k = k P n ( u ( j ) i + v ) k ≥ k P n u ( j ) i k − k P n v k ≥ 2 − ( j +2) − 2 − ( j +3) ≥ 1 8 k x k .  Applying this lemma to X − X there are subspaces U k with dim U k ≤ d k := c (1 + k ) α + β +1 such that k P k z k ≥ k z k / 8 for all z ∈ X − X with k z k ≥ 2 − k . Let S k denote the closed unit ball in U k ; clearly any φ ∈ S k induces a linear functional L φ on H v ia the definition L φ ( u ) = ( φ, u ), where ( · , · ) is the inner pro duct in H . L et ζ > 0 b e fixed and define C ζ = 1 / P ∞ k =1 k − 1 − ζ . W e now define the prob e set (5.2) Q = ( ( l 1 , . . . , l N ) : l n = L φ n where φ n = C ζ ∞ X k =1 k − 1 − ζ φ nk with φ nk ∈ S k ) . 10 ERIC J. OLSON AND JAMES C. ROBINSON W e can identify S j with the unit ball B d j in R d j , and we denote by λ j the pro babilit y measure on S j that cor resp onds to the unifor m probability measure on B d j . W e let µ b e the pr obability measure on Q that results from choos ing ea ch φ nk randomly with resp ect to λ d k . Note that Q is a compact subset of V , and that all ele men ts of Q hav e L ipschit z constant a t most √ N . Before proving our main theo rem w e will prov e a key estimate o n µ . Although the arg umen t is essentially the same as that in Hunt & K aloshin (1999 ) our version is a little more explicit a nd we include it here for c ompleteness. The es timate relies on the following s imple inequality . Lemma 5. 4. If x ∈ R j and η ∈ R then λ j { ω ∈ B j : | η + ( ω · x ) | < ǫ } ≤ cj 1 / 2 ǫ | x | − 1 . wher e c is a c onstant that do es not dep end on η or j . Pr o of. Le t ˆ x = x/ | x | . This follows immediately from estimate λ j { ω ∈ B j : | η + ( ω · x ) | < ǫ } ≤ λ j  ω ∈ B j : | ω · ˆ x | < ǫ | x | − 1  = Ω j − 1 Ω j 2 Z min( ǫ | x | − 1 , 1) 0 (1 − ξ 2 ) ( j − 1) / 2 d ξ ≤ Ω j − 1 Ω j 2 ǫ | x | − 1 where Ω j = π j / 2 Γ( j / 2 + 1) is the volume of the unit ball in R j .  Lemma 5. 5. If x ∈ H and f ∈ V then µ { L ∈ Q : | ( L − f )( x ) | < ǫ } ≤ c ( d 1 / 2 k k 1+ ζ ǫ k P k x k − 1 ) N for every k ∈ N wher e c is a c onstant indep endent of f and k . Pr o of. Given k ∈ N , let J be the index set J = N \ { k } and define B =  M j ∈J B d j  N . Given α = (( α nj ) j ∈J ) N n =1 ∈ B fixed, define A α =  ( φ nk ) N n =1 : | ( η n + k − 1 − ζ φ nk )( x ) | < ǫ for all n  where η n ( x ) = C ζ X j ∈J j − 1 − ζ α nj ( x ) − f n ( x ) . By Lemma 5.4 there is a constant c indep endent o f α , v and k such that λ N d k ( A α ) ≤ c ( d 1 / 2 k k 1+ ζ ǫ k P k x k − 1 ) N . Let P = µ { L ∈ Q : | ( L − v )( x ) | < ǫ } . Then P ≤ µ { L ∈ Q : | ( l n − f n )( x ) | < ǫ for a ll n } . Let Φ N = (  ( φ nk ) ∞ k =1  N n =1 : C ζ    ∞ X k =1 k − 1 − ζ ( φ nk − f n )( x )    < ǫ, ∀ n = 1 , . . . , N ) ALMOST BI-LIPSCHITZ EMBEDDINGS 11 Then b y F ubini’s theorem P ≤  ∞ O j =1 λ d j  N Φ N = Z α ∈ B Z φ ∈ A α d λ N d k ( φ ) d  O j ∈J λ d j  N ( α ) ≤ Z α ∈ B c ( d 1 / 2 k k 1+ ζ ǫ k P k x k − 1 ) N d  O j ∈J λ d j  N ( α ) = c ( d 1 / 2 k k 1+ ζ ǫ k P k x k − 1 ) N . This finishes the pr o of.  5.2. Almost bi-Lipschitz embed di ngs. W e are now in a po sition to state and prov e our ma in theorem, that a compact subset X of a Hilbert spa ce with X − X almo st homogene ous a dmits almo st bi-Lipschitz linear em beddings into finite- dimensional spaces. Unfortunately homoge neit y of X is not automa tically inher ited by X − X : Olson (2002) e xhibits an exa mple of a set X with d A ( X ) = 0 but d A ( X − X ) = + ∞ (for mo re see Section 8). Theorem 5. 6 . L et X b e a c omp act subset of a H ilb ert sp ac e H su ch that X − X is ( α, β ) -almost homo gene ous with d α,β A ( X − X ) < s < N . If γ > 2 + N (3 + α + β ) + 2( α + β ) 2( N − s ) then a pr evalent set of line ar maps f : H → R N ar e inje ctive on X and, in p artic- ular, γ -almost bi-Lipschitz. Pr o of. Fir st c ho ose ζ > 0 in the definition of Q small enough such that (5.3) γ > 2 + N (3 + 2 ζ + α + β ) + 2( α + β ) 2( N − s ) . Since τ ( X ) ≤ d F ( X ) ≤ d F ( X − X ) ≤ d α,β A ( X − X ) we can apply Hunt & Kaloshin’s result (Theorem 5.2, ab ov e) with θ chosen so that θ > N (1 + s/ 2 ) N − s . to obtain a prev ale n t set S 0 of linear functions f : H → R N such tha t f ∈ S 0 implies there exists a θ < 1 a nd c 1 > 0 such that (5.4) | f ( x ) − f ( y ) | ≥ c 1 k x − y k θ for all x, y ∈ X . (W e note here that the co mpactly s uppo rted pr obability measure used in the def- inition of prev alence for S 0 differs from the measure µ constructed in Section 5.1, but is de fined on the same normed linear space V of linear maps from H to R N ). W e use this r esult to b o ots trap a refined argument that makes use of the strong er hypothesis that d α,β A ( X − X ) < ∞ . Let S 1 be the s ubset of V consisting of those linear functions f : H → R N such that f ∈ S 1 implies there exists δ > 0 such that (5.5) | f ( x ) − f ( y ) | ≥ k x − y k slog( k x − y k ) γ for all k x − y k < δ. 12 ERIC J. OLSON AND JAMES C. ROBINSON W e now show t hat the set S 1 is also pre v alent. Given f ∈ V , let K b e the Lipschitz constant o f f . W e wish to show that µ ( f + S 1 ) = 1. This is eq uiv alent to showing that µ ( Q \ ( f + S 1 )) = 0. Define the lay ers of X − X b y (5.6) Z j = n z ∈ X − X : 2 − ( j +1) ≤ k z k ≤ 2 − j o and the set Q j of linear maps that fail to sa tisfy the required contin uit y prop erty 3 for some z ∈ Z j by Q j =  L ∈ Q : | ( L − f )( z ) | ≤ Ψ − γ (2 − j ) for some z ∈ Z j  , where Ψ − γ (2 − j ) := 2 − j σ γ slog(2 − j ) γ and σ is the constant o ccurr ing in (p4) in Lemma 2.1. W e now bound µ ( Q j ). By ass umption d α,β A ( X − X ) < s , and so Z j can b e cov ered b y (5.7) M j ≤ M slog (2 − j ) γ s slog(2 − j ) β slog(Ψ − γ (2 − j )) α ≤ c 2 (1 + j ) α + β + γ s balls of radius Ψ − γ (2 − j ). Let the centres of these ba lls be z ( j ) i ∈ Z j where i = 1 , . . . , M j . Given any z ∈ Z j there is z ( j ) i such that k z − z ( j ) i k ≤ Ψ − γ (2 − j ). Thus | ( L − f )( z ) | ≥ | ( L − f )( z ( j ) i ) | − | ( L − f )( z − z ( j ) i ) | ≥ | ( L − f )( z ( j ) i ) | − ( K + √ N )Ψ − γ (2 − j ) implies Q j ⊆ M j [ i =1 n L ∈ Q : | ( L − f )( z ( j ) i ) | ≤ ( K + 2 √ N )Ψ − γ (2 − j ) o . It follows, setting k = j in Lemma 5.5, that µ ( Q j ) ≤ M j X i =1 µ n L ∈ Q : | ( L − f )( z ( j ) i ) | ≤ ( K + 2 √ N )Ψ − γ (2 − j ) o ≤ M j  d 1 / 2 j j 1+ ζ ( K + 2 √ N )Ψ − γ (2 − j ) k P j ( z ( j ) i ) k − 1  N . Now (5.7) and Lemma 5.3 imply that µ ( Q j ) ≤ c 2 (1 + j ) α + β + γ s  d 1 / 2 j j 1+ ζ ( K + 2 √ N )2 j +3 Ψ − γ (2 − j )  N . In particular (recall that d j ≤ C (1 + j ) α + β +1 ) there is a constant c 3 > 0 independent of j such tha t µ ( Q j ) ∼ c 3 j α + β + γ s + N ( α + β +3+2 ζ − 2 γ ) / 2 as j → ∞ . Since (5.3) implies N (2 γ − 3 − 2 ζ − ( α + β )) / 2 > 1 + α + β + γ s , we hav e ∞ X j =1 µ ( Q j ) < c 4 . It follows fro m the Borel-Cantelli Lemma that µ -almo st every L is con tained in only a finite num b er o f the Q j ; i.e. there exists a J such that for a ll j ≥ J , 3 Strictly speaking the union of the Q j form a set strictly larger than the complemen t of S 1 . ALMOST BI-LIPSCHITZ EMBEDDINGS 13 2 − ( j +1) ≤ k z k ≤ 2 − j implies that | ( L − f )( z ) | ≥ Ψ − γ (2 − j ). It follows fro m (p4) in Lemma 2.1 that | ( L − f )( z ) | ≥ σ γ Ψ − γ ( k z k ) = k z k slog( k z k ) γ for every k z k ≤ 2 − J . Thu s L − f ∈ S 1 and so L ∈ S 1 + f for µ -almost every L . Define S = S 0 ∩ S 1 . Since the intersection of prev alent sets is prev alen t (F a ct 3 ′ in Hunt et al. (1992)) S is pr ev alent. L et f ∈ S . Then ther e is c 1 and δ such that bo th (5.4) and (5.5) ho ld. Thus | f ( x ) − f ( y ) | ≥ c 5 k x − y k slog( k x − y k ) γ for all x, y ∈ X where c 5 = min { 1 , c 1 δ / Ψ − γ ( R ) } a nd R > 0 is s uc h that X − X ⊆ B (0 , R ).  Note that for a s pace X with X − X ho mogeneous, i.e. α = β = 0 in the ab ov e theorem, fo r an y γ > 3 / 2 w e c an c hoo se N large enough to o btain a γ -a lmost bi-Lipschitz embedding into R N . W e will pr ov e a Ba nach space version o f Theo rem 5 .6 in Section 7. How ev er, we delay this while, in the next sectio n, w e consider in more detail almost homogeneity in a Hilb ert space . 6. Lipschitz appro xima ting dimension of Hilber t subsets and H ¨ older-Lipschitz embeddings The stro ng res ult o f the previous se ction requir es that X − X is almost homog e- neous, while for a g eneral almost homo geneous metric space ( X , d ) the embedding result of Theor em 2.3 o nly provides a subset f ( X ) of a Hilb ert space that is itself almost homogeneo us. Here w e in v estigate f urther some of the prop erties of f ( X ), and are lea d to define the ‘Lipschitz a pproximating dimension’ and the ‘Lipsc hitz deviatio n’. In particular we show that it is p os sible to replace Hunt & Kalo shin’s thickness exp onent with the Lipschitz dev iation. 6.1. F urther prop erties o f the image f ( X ) . First we consider the almo st bi- Lipschitz ima ge f ( X ) of a compact a lmost homog eneous metric space ( X , d ) in a Hilber t space , as provided by Theo rem 2.3. W e show tha t f ( X ) can b e very well approximated by linear subspaces: it has ‘better than zer o’ thic kness. As remarked after the proo f of The orem 2.3, when ( X, d ) is compa ct the function f defined by the simplified serie s f ( x ) = ∞ X j = − k 2 − j (1 + | j | ) δ M j φ j ( x ) ⊗ e j still provides a γ -almo st bi-Lipsc hitz embedding of X into a H ilber t space (c hoosing a k suc h that d ( x 1 , x 2 ) ≤ 2 k for all x 1 , x 2 ∈ X ). No w, for n ∈ N any element o f f ( X ) can b e a pproximated to within B ∞ X j = n +1 2 − j (1 + | j | ) δ ≤ B ∞ X j = n +1 2 − j ≤ B 2 − n 14 ERIC J. OLSON AND JAMES C. ROBINSON by an element of the subspace U = n M j = − k R M j ⊗ e j , which ha s dimension (6.1) n X j = − k M j ≤ ( n + k + 1) C (1 + n ) α + β ≤ c 1 (1 + n ) α + β +1 . Here c 1 depe nds on C , k and the co nstants in Lemma 2.1 but is indep endent of n . It follows that (6.2) d ( f ( X ) , ǫ ) ≤ c 2  log(e + 1 / ǫ )  α + β +1 . One cons equence of this inequality is that the thickness exp onent of f ( X ) is zero, but (6.2) is sig nificantly stro nger than this. 6.2. The Lipschitz deviatio n . Inspired b y the quan tit y d ( X, ǫ ) us ed to define the thickness w e now in tr o duce a mor e general quantit y , the m -Lips ch itz deviation: we denote by δ m ( X, ǫ ) the smallest dimension o f a linear subspace U suc h that dist( X , G U [ φ ]) < ǫ for some m -Lipschitz function φ : U → U ⊥ , k φ ( u ) − φ ( v ) k ≤ m k u − v k for all u, v ∈ U, where U ⊥ is the orthogona l complement of U in H . W e will wr ite G U [ φ ] for the graph of φ over U : G U [ φ ] = { u + φ ( u ) : u ∈ U } . Clearly δ 0 ( X, ǫ ) = d ( X , ǫ ). In Sec tion 6 .1 we showed that for the almost bi- Lipschit z embedding f ( X ) o f an almost homogeneo us metric space into a Hilbert space δ 0 ( f ( X ) , ǫ ) ≤ c 2  log(e + 1 / ǫ )  α + β +1 . W e now show that Lemma 5.3 implies a b ound of a similar for m on δ 8 ( X, ǫ ) for any subset of a Hilb ert spa ce with X − X almost homog eneous. Prop ositio n 6. 1. L et X b e a c omp act su bset wi th t he set of differ en c es X − X ( α, β ) -almost ( M , s ) -homo gene ous. Then ther e exists a se quenc e of line ar subsp ac es U k with dim U k ≤ C (1 + k ) α + β +1 and U k +1 ⊇ U k , and 8 -Lipschitz functions φ k : U k → U ⊥ k such that dist( X, G U k [ φ k ]) ≤ 2 − k . In p articular δ 8 ( X, ǫ ) ≤ K  log(e + 1 / ǫ )  α + β +1 . Pr o of. Applying Lemma 5 .3 to X − X we obtain a nested se quence o f linear sub- spaces for which 1 8 k x − y k ≤ k P k x − P k y k ≤ k x − y k for all x, y ∈ X with k x − y k ≥ 2 − k , where P k is the or thogonal pro jection onto U k . Define φ k : U k → U ⊥ k as follows. Let N k be a maximal 2 − k net in ( X , d ) and se t φ k ( P k x ) = ( I − P k ) x for x ∈ N k . Given P k x, P k y ∈ P k N k we have k φ k ( P k x ) − φ k ( P k y ) k ≤ k ( I − P k )( x − y ) k ≤ k x − y k ≤ 8 k P k x − P k y k . ALMOST BI-LIPSCHITZ EMBEDDINGS 15 Therefore φ k : P k N k → U ⊥ k is a 8- Lipschitz function. Now, extend this φ k to a 8-Lipschitz function U k → U ⊥ k . Since N k ⊂ G U k [ φ k ] then any p oint of X lies within 2 − k of G U k [ φ k ]. Thus δ 8 ( X, 2 − k ) ≤ c 2 (1 + k ) α + β +1 and the result follows.  W e now show that this arg umen t can b e rev ersed, i.e. tha t the results of Lemma 5.3 and Pro po sition 6.1 are essentially e quiv alent. Prop ositio n 6.2. Supp ose that X is a c omp act subset of a Hilb ert sp ac e X . F or any m ≥ 0 let { U k } ∞ k =1 b e a se qu enc e of line ar su bsp ac es such t hat for e ach U k ther e exists an m -Lipschitz function φ k : U k → U ⊥ k with dist( X, G U k [ φ k ]) ≤ 2 − k . Then ther e exists an inte ger n and a c onstant c m > 0 (which dep ends on m but is indep endent of k ) su ch t hat for every k k P k + n ( x 1 − x 2 ) k ≥ c m k x 1 − x 2 k for all x, y ∈ X with k x 1 − x 2 k ≥ 2 − k . Pr o of. Fir st note that for a n y x ∈ H we hav e dist( x, G U k [ φ k ]) 2 = inf u ∈ U k  k P k x − u k 2 + k ( I − P k ) x − φ k ( u ) k 2  and since for a n y u ∈ U k we have k ( I − P k ) x − φ k ( P k x ) k 2 = k ( I − P k ) x − φ k ( u ) + φ k ( u ) − φ k ( P k x ) k 2 ≤ 2 k ( I − P k ) x − φ k ( u ) k 2 + 2 k φ k ( u ) − φ k ( P k x ) k 2 ≤ 2 k ( I − P k ) x − φ k ( u ) k 2 + 2 m 2 k u − P k x k 2 ≤ l 2 m  k P k x − u k 2 + k ( I − P k ) x − φ k ( u ) k 2  , where l 2 m = 2 max(1 , m 2 ), it follows that for x ∈ X (6.3) k ( I − P k ) x − φ k ( P k x ) k ≤ l m dist( x, G U k [ φ k ]) ≤ l m 2 − k Now supp ose that x 1 , x 2 ∈ X with k x 1 − x 2 k ≥ 2 − k . Let n b e the sma llest in teger such tha t 3 l m ≤ 2 n and set ˜ x j = P k + n x j + φ k + n ( P k + n x j ) for j = 1 , 2 . Clearly , P k + n ( x 1 − x 2 ) = P k + n ( ˜ x 1 − ˜ x 2 ). F urther more, it follows fr om (6.3) tha t | x j − ˜ x j | ≤ 2 − k / 3 for j = 1 , 2. Therefore , | ˜ x 1 − ˜ x 2 | ≥ | x 1 − x 2 | / 3. Now, since ˜ x 1 , ˜ x 2 ∈ G U k + n [ φ k + n ], k P k + n ˜ x 1 − P k + n ˜ x 2 k 2 = k ˜ x 1 − ˜ x 2 k 2 − k φ k + n ( P k + n ˜ x 1 ) − φ k + n ( P k + n ˜ x 2 ) k 2 ≥ k ˜ x 1 − ˜ x 2 k 2 − m 2 k P k + n ( ˜ x 1 − ˜ x 2 ) k 2 , and so k P k + n ( x 1 − x 2 ) k = k P k + n ( ˜ x 1 − ˜ x 2 ) k ≥ k ˜ x 1 − ˜ x 2 k √ 1 + m 2 ≥ k x 1 − x 2 k 3 √ 1 + m 2 .  16 ERIC J. OLSON AND JAMES C. ROBINSON 6.3. Almost homogeneo us subsets of a Hil b ert space. If we assume o nly the almost ho mogeneity of X , rather than of X − X , we ca n a pply a simplified v aria n t of the argument of Theor em 5.6 to o btain the following minor improvemen t to the embedding theorem of Hun t & Kalo shin (under o ur stronger h ypothes is). F o r a zero thickness set X with d F ( X ) ≤ d they obtain a n upp er limit of N / ( N − 2 d ) for the H¨ older exp onent, while under the a ssumption tha t d α,β A ( X ) ≤ s we obtain ( N − s ) / ( N − 2 s ) as the upp er limit. Note that w e replace an y assumption o n the thickness b y (6.4 ), which in particular is s atisfied by the a lmost bi- Lipschitz embedding f ( X ) of an almost homogeneo us metric space with m = 0 (see 6.2). Theorem 6.3. Supp ose that X is a c omp act subset of a Hilb ert sp ac e H with d α,β A ( X ) < s and that for some m > 0 , σ ≥ 0 , (6.4) δ m ( X, ǫ ) ≤ K [log(e + 1 / ǫ )] σ . Then for any inte ger N > 2 s , if θ > ( N − s ) / ( N − 2 s ) t her e is a pr evalent set S of line ar maps f : H → R N such that for every f ∈ S t her e exists c > 0 such that (6.5) | f ( x ) − f ( y ) | ≥ c k x − y k θ for al l x, y ∈ X. Pr o of. Set d j = δ m ( X, 2 − j ) ≤ K  log(e + 2 j )  σ and define Q as in (5 .2) with ζ = 1. Define the lay er s Z j as in (5.6) and Q j =  L ∈ Q : | ( L − v )( z ) | ≤ 2 − j θ for some z ∈ Z j  . Let R > 0 be chosen so large that X ⊂ B (0 , R ). Cov er X by N X ( R, 2 − ( j +1) θ ) ≤ M  R 2 − ( j +1) θ  s slog( R ) β slog(2 − ( j +1) θ ) α ≤ c 1 2 j θ s (1 + j θ ) α balls of radius 2 − ( j +1) θ centred a t points x i ∈ X . Denote these as X i = n x ∈ X : k x − x i k < 2 − ( j +1) θ o . Now cons ider the larger balls B i = n y ∈ X : k x i − y k ≤ 2 − ( j +1) θ + 2 − j o . Cov er ea ch o f these balls by at most N X (2 − ( j +1) θ + 2 − j , 2 − ( j +1) θ ) ≤ M  1 + 2 ( j +1) θ − j  s slog(2 − ( j +1) θ + 2 − j ) β slog(2 − ( j +1) θ ) α ≤ c 2 2 j ( θ − 1) s (1 + j ) β (1 + j θ ) α balls of radius 2 − ( j +1) θ . Since Z j = [ i [ x ∈ X i n x − y : 2 − ( j +1) < k x − y k < 2 − j o ⊆ [ i ( X i − B i ) it follows that Z j can b e covered by M j = c 1 c 2 2 j s (2 θ − 1) (1 + j θ ) 2 α (1 + j ) β balls of radius 2 − j θ . Le t z ( j ) i denote the centres of these balls. ALMOST BI-LIPSCHITZ EMBEDDINGS 17 Applying similar estimates a s in the pro of of The orem 5 .6 (these rely on Prop o- sition 6.2 to ensure that k P k z ( j ) i k ≥ c k z ( j ) i k for some c > 0) shows that µ ( Q j ) ∼ 2 j s (2 θ − 1) j 2 α + β [ j 2+ σ 2 j (1 − θ ) ] N as j → ∞ . Thu s P µ ( Q j ) conv erges provided that θ > ( N − s ) / ( N − 2 s ). The argument is now co ncluded as in Theorem 5.6.  By combining this with Prop ositio n 2.3 w e obtain the following H¨ older-Lipschitz embedding result for homog eneous metric spaces (cf. Lemma 9.1 in F oias and O l- son (1996) whic h has a similar result for spaces with finite upper b ox-counting dimension). Corollary 6.4. L et ( X , d ) b e an almost homo gene ous metric sp ac e with d α,β A < s . If N > 2 s and θ > ( N − s ) / ( N − 2 s ) ther e exists a map φ : ( X , d ) → R N such t hat c − 1 d ( x, y ) θ ≤ | φ ( x ) − φ ( y ) | ≤ c d ( x, y ) for al l x, y ∈ X . Of course one ca n pro ve finite-dimensiona l v ersions of Theorems 5.6 and 6.3 using very simila r techniques. 6.4. The Lipsc hitz deviation. It is in teresting that our argument sho ws that fo r any fixed m > 0 the thic kness exp onent in the statement of Theor em 5.2 c an be replaced by the m -Lipschitz deviation , de v m ( X ), which w e define by analogy with the thickness exponent (cf. (5.1)) dev m ( X ) = lim sup ǫ → 0 log δ m ( X, ǫ ) − log ǫ . W e note that dev m ( X ) ≤ τ ( X ) and that this g ives an indication of why the thic k- ness exp onent can b e ex pec ted to play a r ˆ ole in determining the H¨ older exp onent in (6.5). W e state without pr o of: Theorem 6.5. L et X b e a c omp act subset of a Hilb ert sp ac e H , D an inte ger with D > d F ( X − X ) , and let dev m ( X ) b e the m -Lipschitz deviation of X . If θ is chosen with θ > D (1 + dev m ( X ) / 2) D − d F ( X − X ) then for a pr evalent set of line ar maps L : B → R D ther e exist s a c > 0 such that c k x − y k θ ≤ | Lx − L y | ≤ k L k k x − y k for al l x, y ∈ X ; in p articular these maps ar e inje ctive on X . 7. Embedding subsets X of Banach sp aces with X − X homogeneous In this se ction we e xtend the Hilb ert s pace result to c ov e r subsets of Banach spaces. In particular this enables us to prov e a new almo st bi-L ipschit z embedding result for a class o f metric spaces. The k ey p oint is, of course, tha t eno ugh of Lemma 5.3 can be salv a ged to follow a very simila r pro of: Lemma 7.1. L et X b e an ( α, β ) -almost ( M , s ) -homo gene ous subset of a Banach sp ac e B . Then ther e exists a neste d se quenc e of subsets U n +1 ⊇ U n such that dim U n ≤ C (1 + n ) α + β +1 18 ERIC J. OLSON AND JAMES C. ROBINSON and dist( x, U n ) ≤ 1 4 k x k fo r al l k x k ≥ 2 − n In particula r, if we apply this lemma to Z = X − X , there exists a ne sted sequence of linear subspaces of B , U k ⊆ U k +1 such that given z ∈ X − X with k z k ≥ 2 − n there exists a p oint ˜ z ∈ U n such that k z − ˜ z k ≤ 1 4 k z k and k ˜ z k ≥ 3 4 k z k . W e now let S k denote the clos ed unit ball in the dual of U k , and denote b y S E k an isometric embedding of S k int o B ∗ , whose existence is guara n teed b y the Hahn-Banach theor em. W e then define our prob e set Q as Q = ( ( l 1 , . . . , l N ) : l n = C ζ ∞ X k =1 k − 1 − ζ φ nk with φ nk ∈ S E k ) . Cho osing a basis for S k we identify S k with a conv ex set U k ⊂ R d k , and induce a probability measur e on S k (and hence on S E k ) via the unifor m pr obability mea sure on U k . W e now outline the pro o f of the following r esult: Theorem 7.2. L et X b e a c omp act subset of a Hilb ert s p ac e B such that X − X is ( α, β ) -almost homo gene ous with d α,β A ( X − X ) < s < N . If γ > 1 + N (2 + α + β ) + ( α + β ) N − s then a pr evalent set of line ar maps f : B → R N ar e inje ctive on X and, in p artic- ular, γ -almost bi-Lipschitz. Pr o of. The pro of pr o ceeds identically to that of Theor em 5.6 until we ha ve to estimate µ  L : | ( L − f ) z ( j ) i | ≤ c Ψ − γ (2 − j )  . W e can now follow the arg umen t from Hun t & K aloshin (19 99), with so me small changes—w e only highlight these here. In our c ase we know that there exists a po int ˜ z ( j ) i ∈ U j such that k ˜ z ( j ) i − z ( j ) i k ≤ 1 4 k z ( j ) i k . It follows that there exists a ψ ∈ S j such that ψ ( z ( j ) i ) ≥ k z ( j ) i k − k z ( j ) i − ˜ z ( j ) i k ≥ 3 4 k z ( j ) i k ≥ 3 · 2 − ( j +3) . W e ca n then follow Hunt & Kaloshin’s arg umen t to sho w that µ  L : | ( L − f ) z ( j ) i | ≤ c Ψ − γ (2 − j )  ≤  j 1+ ζ d j 2 j +3 Ψ − γ (2 − j )  N , and the pro o f is completed exa ctly as in the Hilbert spa ce cas e, noting that we now hav e a factor of d j rather than o nly d 1 / 2 j .  One sig nificant consequence o f extending the result to Banach spaces is it allows for a new result for metric spac es via the Kuratowski isometric em bedding o f ( X, d ) int o L ∞ ( X ): cho osing an arbitrar y point x 0 ∈ X , this is given b y (7.1) x 7→ ρ x , where ρ x ( y ) = d ( x, y ) − d ( x 0 , y ) ALMOST BI-LIPSCHITZ EMBEDDINGS 19 (see Heinonen, 2003, for example). In this wa y we can attach meaning to X − X for an arbitra ry metric space ( X , d ), i.e. (7.2) X − X = { f ∈ L ∞ ( X ) : f = d ( x, · ) − d ( y , · ) , x, y ∈ X } . W e then have the following result: Theorem 7.3 . L et ( X, d ) b e a c omp act metric sp ac e such that X − X is an almost homo gene ous subset of L ∞ ( X ) . T hen ther e exists an inje ctive almost bi-Lipschitz map f : ( X, d ) → R N . Pr o of. Denote by F : ( X , d ) → L ∞ ( X ) the isometric embedding in (7.1). Then F ( X ) is isometric to ( X , d ), while the set of differences F ( X ) − F ( X ) is almost homogeneous by assumption. The existence of an injectiv e almost bi-Lipschitz embedding of F ( X ) into R N , which follows from the Ba nach spa ce version of o ur main theorem, immediately implies the existence of the same t ype of embedding for ( X , d ) into R N .  8. The rela tionship between d α,β A ( X ) and d α,β A ( X − X ) In this section we give some results relating the homoge neit y of X a nd X − X . First, we giv e an example of a set X for which d A ( X ) = 0 but d A ( X − X ) = + ∞ . It is easy to show that the set (8.1) X ∗ = n a n e n : a n = 4 − (2 j ) , n = 2 j − 1 , . . . , 2 j − 1 o , where e n is an or thonormal basis of a Hilb ert space H , has d A ( X ∗ ) = + ∞ . Note that | a n | ≤ 4 − n for all n . Consider no w the subset X of H × H defined by X =  (4 − n e n , a n e n )  ∞ n =1 ∪  (4 − n e n , 0)  . A simple argument shows that d A ( X ) = 0, while X − X contains a cop y of X ∗ and so d A ( X − X ) = ∞ . This nega tive result app ears to b e in some wa ys typical for a lmost homo geneous sets as well, a s we will now show. W e b egin with a pr eparator y lemma. Lemma 8. 1. The ortho gonal s e quenc e with algebr aic de c ay X ∗ =  b n e n : b n ∼ ǫn − γ  wher e ǫ, γ > 0 has d α,β A ( X ∗ ) = + ∞ for any α, β ≥ 0 . Pr o of. Le t n 0 be chosen so large that ǫ (2 n ) − γ < | b n | < ǫ ( n/ 2) − γ for n > n 0 . Let r n = ǫ ( n/ 2) − γ and ρ n = ǫ (4 n ) − γ . Supp ose, for a c ontradiction, that d A ( X ∗ ) < s < ∞ . Then there exists an M ≥ 1 such that (8.2) N ( r n , ρ n ) ≤ M  r n ρ n  s slog( r n ) β slog( ρ n ) α . On the other hand, B (0 , r n ) ⊇ { b k e k : n < k ≤ 2 n } , where the points b k e k with n < k ≤ 2 n are each a distance greater than | b k | > ǫ (4 n ) − γ apart from each other. There fore, (8.3) N ( r n , ρ n ) ≥ c ard  { b k e k : n < k ≤ 2 n }  = n. 20 ERIC J. OLSON AND JAMES C. ROBINSON Combining ineq ualit y (8.2) with (8.3) a nd applying (p1) of Lemma 2 .1 we obta in n ≤ M 8 γ s  log 2 + | lo g ǫ ( n/ 2) − γ |  β  log 2 + | lo g ǫ (4 n ) − γ |  β . Letting n → ∞ yields a contradiction, and so d α,β A ( X ∗ ) = ∞ .  Lemma 8. 2. Given two un it ve ctors v , w ∈ H set e 1 = v and cho ose α ∈ R and a unit ve ctor e 2 such that e 1 cos α − e 2 sin α = w and co s α = ( v , w ) . Note that e 2 is ortho gonal to e 1 . Extend { e 1 , e 2 } to a b asis for H , and define the r otation R x =  cos( αψ ( x )) sin( αψ ( x )) − sin( αψ ( x )) cos( αψ ( x ))  ⊕ id , wher e ψ : H → R is a fi xe d C ∞ function su ch that ψ ( x ) =  0 if k x k ≤ 3 / 4 or k x k ≥ 2 , 1 if k x k = 1 . L et f ( x ) = R x x . Then f ∈ C ∞ and f ( v ) = w . Mor e over, f η ( x ) = η − 1 f ( ηx ) is uniformly bi-Lipschitz c ontinuous for η > 0 and differ ent fr om t he identity only for x ∈ H s uch that (3 / 4) η − 1 < k x k < 2 η − 1 . Pr o of. By constr uction f ∈ C ∞ , f ( v ) = w and f ( x ) = x for k x k ≤ 3 / 4 or k x k ≥ 2. Rescaling shows that f η ( x ) is different from the iden tit y only for (3 / 4) η − 1 < k x k < 2 η − 1 . W e now sho w that f η ( x ) is uniformly bi-Lipschitz co n tin uous for η > 0. Let x, y ∈ H with k x k ≤ k y k . If k x k ≥ 2 η − 1 then f η ( x ) = x and f η ( y ) = y , so we cons ider only the case k x k < 2 η − 1 . Then k f η ( x ) − f η ( y ) k = k R ηx x − R ηy y k ≤ k ( R ηx − R ηy ) x k + k R ηy ( x − y ) k ≤ k R ηx − R ηy kk x k + k R ηy kk x − y k ≤ 2 η − 1 k R ηx − R ηy k + k x − y k . Since k R ηx − R ηy k =      cos( αψ ( η x )) − co s( αψ ( η y )) sin( αψ ( η x )) − sin( αψ ( η y )) − sin( αψ ( η x )) + sin( αψ ( η y )) co s( αψ ( η x )) − cos( αψ ( η y ))      ≤ C 1 αη k x − y k := C 2 η k x − y k , it follows that k f η ( x ) − f η ( y ) k ≤ (2 C 2 + 1 ) k x − y k where the Lipschitz co nstant 2 C 2 + 1 do es no t dep end on η . Since f η is injective with in v erse f − 1 η formed by the sa me cons truction but with the roles of v a nd w reversed we obtain the sa me bo und for k f − 1 η ( x ) − f − 1 η ( y ) k .  Prop ositio n 8.3. L et X b e a c onne cte d su bset of a Hilb ert sp ac e H that c ontains mor e than one p oint. Th en t her e exists a C ∞ bi-Lipschitz map φ : H → H such that d α,β A ( φ ( X ) − φ ( X )) = + ∞ for every α, β ≥ 0 . F u rt hermor e φ may b e chosen such that dis t H ( φ ( X ) , X ) is arbitr arily smal l. ALMOST BI-LIPSCHITZ EMBEDDINGS 21 Pr o of. Since X con tains mor e than one p oint, there exist t wo disjo in t balls B ( x 1 , R ) and B ( x 2 , R ) of radius R > 0 . Moreover, since X is connected, then there are points x 2+ i ∈ X for i = 1 , 2 s uc h tha t k x 2+ i − x i k = R / 4. Thus, the four ba lls B ( x i , R/ 8) with x i ∈ X for i = 1 , . . . , 4 are disjoint. Moreover 4 [ i =1 B ( x i , R/ 8) ⊆ 2 [ i =1 B ( x i , 3 R / 8) . Recursively define nested families of dis joint ba lls such that 2 j +1 [ i =1 B ( x i , R 8 − j ) ⊆ 2 j [ i =1 B ( x i , 3 R 8 − j ) . F or j = 0 , 1 , 2 , . . . and i = 1 , . . . , 2 j +1 let a j = (1 / 2 ) R 8 − j and e ij = e 2 j +1 − 2+ i where e i is an or thonormal basis of H . Cho ose the p oints y ij ∈ B ( x i , R 8 − j ) such that k x i − y ij k = a j . F urther define g ij ( x ) = x i + f η ( x − x i ) , where f η is the function given in Lemma 8 .2 for v = ( y ij − x i ) /a j , w = e ij and η = 1 /a j . If k x − x i k ≥ 2 a j = R 8 − j or k x − x i k ≤ (3 / 4) a j = 3 R 8 − j − 1 then f η ( x − x i ) = x − x i and g ij ( x ) = x . Therefore the function g ij is C ∞ , bi-Lipschitz and different from the identit y only on the annulus B ( x i , R 8 − j ) \ B ( x i , 3 R 8 − j − 1 ). Moreov er, by construction we ha v e g ij ( y ij ) = x i + f η ( y ij − x i ) = x i + a i f ( v ) = x i + a i e ij . Set φ ( x ) = ∞ X j =0 2 j +1 X i =1 g ij ( x ) . Since t he g ij are diff erent from the identit y o nly on disjoint sets and the bi-Lipschitz constant o f f η is independent of η , then the map φ is a bi-Lipschitz C ∞ map of H onto H . Since φ ( X ) − φ ( X ) contains  a j e ij : j = 0 , 1 , 2 , . . . and i = 1 , . . . , 2 j +1  =  b n e n : b n = (1 / 2) R 8 − j , n = 2 j +1 − 1 , . . . , 2 j +2 − 2  where 4 R / ( n + 2) 3 ≤ b n ≤ 4 R / ( n + 1) 3 , then b n ∼ 4 R n − 3 and hence Lemma 8 .1 implies d α,β A ( φ ( X ) − φ ( X )) = ∞ . Finally , note that dist H ( φ ( X ) , X ) may b e made ar bitrarily sma ll b y taking R > 0 sufficiently small in step one.  A consequence o f this result is that it is not necessary for X − X to b e homoge- neous in order to obtain a bi-Lipschitz embedding o f X in to some R k . Indeed, any set X that can b e so em b edded has a bi-Lipschitz image that ha s d α,β A ( X − X ) = ∞ . How ever, it may still b e the case that X − X has to b e ho mogeneous in order to obtain a line ar bi- Lipschitz em bedding a s in Theorem 5.6. On a mor e p ositive note, if X is an o rthogona l sequence then homogeneity of X do es imply homoge neit y of X − X . Lemma 8.4. L et X = { x j } ∞ j =1 b e an ortho gonal se quenc e in H . If d A ( X ) < + ∞ then d A ( X − X ) ≤ 2 d A ( X ) . 22 ERIC J. OLSON AND JAMES C. ROBINSON Pr o of. Supp ose that X is ( M , s )-ho mogeneous. W e wr ite B X ( r , x ) = B ( r, x ) ∩ X , and co nsider a ball B = B X − X ( r , x − y ) ⊆ X − X o f r adius r centred a t x − y ∈ X − X . Since B ⊆ B X − X ( ρ, 0) ∪  B \ { 0 }  , we ne ed only cov er B \ { 0 } . Suppo se tha t x = y , so that B = B X − X ( r , 0). Let a − b ∈ B \ { 0 } . Then a 6 = b and therefore a is o rthogona l to b . It follows tha t   ( a − b ) − ( x − y )   2 = k a k 2 + k b k 2 < r 2 . Hence a, b ∈ B X ( r , 0), and consequently B \ { 0 } ⊆ B X ( r , 0) − B X ( r , 0) . Cov er B X ( r , 0) with M (2 r /ρ ) s balls B X ( ρ/ 2 , a i ) of r adius ρ/ 2 centred a t a i ∈ X . Then [ i,j B X − X ( ρ, a i − a j ) ⊇ [ i B X ( ρ/ 2 , a i ) − [ j B X ( ρ/ 2 , a j ) ⊇ B X ( r , 0) − B X ( r , 0) ⊇ B X − X ( r , 0) \ { 0 } . It follows that B is cov ered by 1 + M 2 (2 r/ ρ ) 2 s balls of radius ρ . Now supp ose that x 6 = y . Let a − b ∈ B \ { 0 } . Again a 6 = b and therefor e a is orthogo nal to b . W e have k ( a − b ) − ( x − y ) k 2 =    k a − x k 2 + k b − y k 2 k a + y k 2 + k 2 x k 2 k 2 y k 2 + k b + x k 2 if a 6 = y , b 6 = x a 6 = y , b = x a = y , b 6 = x, and so a ∈ B X ( r , x ) b ∈ B X ( r , y ) a ∈ B X ( r , − y ) b ∈ B X ( r , x ) a ∈ B X ( r , y ) b ∈ B X ( r , − x ) a ∈ B X ( r , y ) b ∈ B X ( r , x )        if        a 6 = y , b 6 = x a 6 = y , b = x a = y , b 6 = x, a = y , b = x. Therefore B \ { 0 } ⊆  B X ( r , x ) − B X ( r , y )  ∪  B X ( r , − y ) − B X ( r , x )  ∪  B X ( r , y ) − B X ( r , − x )  ∪  B X ( r , y ) − B X ( r , x )  . Cov er each of B X ( r , x ), B X ( r , − x ), B X ( r , y ) and B X ( r , − y ) by M (2 r/ ρ ) s balls of radius ρ/ 2. An ar gument similar to b efore yields a cov er of B b y 1 + 4 M 2 (2 r/ ρ ) 2 s balls of radius r / 2. Since we hav e N X − X ( r , ρ ) ≤ 1 + 4 M 2 (2 r/ ρ ) 2 s it follows that d A ( X − X ) ≤ 2 s .  9. Non-existence o f bi-Lipschitz linear embeddings In this section we give a simple example sho wing that if w e require a linear embedding (as in Theo rem 5.6 ) then we can do no b etter than almost bi-Lipsc hitz. First we prove the following simple decomp osition lemma for linea r maps from H onto R k (cf. comments in Hun t & K aloshin, 1997). Lemma 9. 1 . Supp ose L : H → R k is a line ar m ap with L ( H ) = R k . Then U = (ker L ) ⊥ has dimension k , and L c an b e de c omp ose d uniquely as M P , wher e P is t he ortho gonal pr oje ct ion onto U and M : U → R k is an invertible line ar map. Note that the r esult of this lemma s hows Theorem 5.6 r emains true with linear maps replaced by or thogonal pro jections. This gives a m uc h more concise pro of of the result in F riz & Robinson (199 9). ALMOST BI-LIPSCHITZ EMBEDDINGS 23 Pr o of. Le t U = (ker L ) ⊥ and s uppo se that there exis t m > k linearly indep endent elements { x j } m j =1 of U for whic h Lx j 6 = 0. Then { Lx j } are elements of R k ; since m > k at lea st one of the { Lx j } can b e written as a linea r co m bination of the others: Lx i = X j 6 = i c j ( Lx j ) . It follows that  x i − X j 6 = i c j x j  = 0 , which c ontradicts the definition of U . Let P denote the o rthogona l pr o jection onto U , a nd M the res triction of L to U . Let x ∈ H , and decomp ose x = u + v , where u ∈ U and v ∈ ker L . Note that this decomp osition is unique. Clea rly Lx = Lu = M u = M ( P x ). It re mains to show that M is invertible. This is clear since dim U = dim R k = k and M is linear.  F ollowing Ben-Artzi et al. (19 93) w e now pr ov e Lemma 9.2. Supp ose that X − X c ontains a s et of the form { α n e n } ∞ n =1 with { e n } ∞ n =1 an orthonormal set. Then no line ar map int o any R k c an b e bi-Lipschitz b etwe en X and its image. Pr o of. W e assume that L ( H ) = R k , other wise it is p oss ible to prune some redun- dant dimensions from R k . Supp ose that L is bi-Lipschitz from X in to R k . W rite L = M P as in Lemma 9.1. Since L is bi- Lipschitz on X then for a ll y ∈ X − X we hav e k y k ≤ c | Ly | = c | M P y | ≤ C k P y k , where C = c k M k . In particular we ha v e k α n e n k ≤ c k P ( α n e n ) k ⇒ c k P e n k ≥ 1 . But k = ra nk P = T rac e P ≥ ∞ X n =1 ( P e n , e n ) = ∞ X n =1 k P e n k 2 = + ∞ a contradiction.  W e no te that this res ult also follows fro m Lemma 2.4 in Mov ahedi-Lank arani & W ells (20 05) which gives a characterisation of sets X that can b e linear ly bi- Lipschitz embedded in to so me R k : such an em bedding is pos sible if and only if the weak clo sure of  x − y k x − y k : x, y ∈ X , x 6 = y  do es not contain ze ro (“weak s pherical compactness of X ” ). Now consider the ho mogeneous set X = { 2 − n e n } ∪ { 0 } , which ha s d A ( X ) = 0. Since X is an orthogonal sequence, it follows that X − X (which in particular contains X ) is also homo geneous; but Lemma 9.2 shows tha t no linea r map in to any finite-dimensional Euclidean space can b e bi-Lipschitz o n X . This sho ws that, with the requir ement of linea rly , our The orem 5.6 cannot b e improved. How ever, note that there is a simple nonline ar bi-Lipschitz map φ from X in to [0 , 1], given b y φ (2 − n e n ) = 2 − n : 24 ERIC J. OLSON AND JAMES C. ROBINSON for n < m we hav e 1 4 (2 − n + 2 − m ) | {z } 1 4 | 2 − n e n − 2 − m e m | ≤ 2 − ( n +1) ≤ | 2 − n − 2 − m | | {z } | φ (2 − n e n ) − φ (2 − m e m ) | ≤ 2 − n ≤ (2 − n + 2 − m ) | {z } | 2 − n e n − 2 − m e m | . The relationship b et w een linear embeddings and general bi-Lipschitz embeddings is delicate. Supp ose that X is a connected set containing more than one po int . The result of Pr op osition 8.3 shows that ev en if X can be linearly bi-Lipsc hitz embedded int o so me R n it is nevertheless bi-Lipschitz equiv alen t to a s pace φ ( X ) that ca nnot be bi-Lipschitz em bedded into an y R n using a linear map. 10. Conclusion W e hav e identified a new cla ss o f almost homo gene ous metric spaces, and s hown that such spaces enjoy almost bi-Lipschitz embeddings into Hilbert space. F urther- more w e have s hown tha t a ny c ompact subset X of a Ba nach s pace w ith X − X almost ho mogeneous ca n em bedded into a finite-dimensional Euclidean s pace is a n almost bi-Lipschitz wa y , and used this to deduce the s ame for an y compact metric space ( X , d ) with F ( X ) − F ( X ) almos t homogeneous, where F : X → L ∞ ( X ) is the isometric Kuratowski em bedding of ( X , d ) into L ∞ ( X ). Some outstanding pr oblems remain: (1) Is there a homogeneous subset of a Hilb ert space that cannot b e bi- Lipschitz embedded into any R k ? (2) Can any (almo st) homogeneo us subse t of a Hilbert space b e (a lmost) bi- Lipschitz em bedded into some R k ? (3) Can one construct an almos t bi-Lipsc hitz em b edding f of a co mpact almost homogeneous metric space ( X, d ) into a Hilb ert space in such a wa y tha t X − X is almo st homogeneous? (This would ans w er (2) po sitively .) (4) Is the exp onent γ in Theor em 5.6 (the p ow er of the slog term) in any w a y optimal? (5) Can one bound the Assouad dimensio n o f the attractors of dissipative PDEs (or preferably the set of differences of solutions lying on such attractors)? References 1. P . Ass ouad Plongements lipschitziens dans R n . Bul letin de l a S. M. F. 111 (1983), 429–448 2. A. Ben-A rtzi, A. Eden, C. F oias, and B. Nicolaenko, H¨ older c ontinuity for the inverse of Ma˜ n ´ e’s pr oje c tion. J. Math. Anal. Appl. 17 8 (1993), 22–29. 3. M.G. Bouligand, Ensumbles Impr opr es et Nombr e Dimensionnel , Bull. Sci. Math., 5 2 (1928), 320–344, 361–376 . 4. C. F oias and E. J. Olson, Finite fr actal dimension and H¨ older-Lipschitz p ar ametrization. Indiana Univ. Math. J. 4 5 (1996), 603–616. 5. P . K. F r iz and J.C. Robinson, Smo oth attr actors have zer o “thick ness”. J. Math. Anal. Appl. 240 (1999), 37–46. 6. P . Ha j lasz, Whitney’s example by way of Assoua d’s emb e dding. Pro c. Amer. Math. So c. 131 (2003), 3463–34 67. 7. J. Heinonen, Geo metric emb e ddings of metric sp ac es Rep ort. University of Jyv¨ askyl¨ a D epart- men t of Mathematics and Statistics, 90. Uni v ersity of Jyv¨ askyl¨ a, Jyv¨ askyl¨ a, 2003 . 8. B.R. Hunt, T. Sauer, and J.A. Y orke, Pr evalenc e: a t ra nslation-invariant almost eve ry for infinite dimensional sp ac es. Bull. Amer. M ath. Soc. 2 7 (1992), 217–238; Pr evalenc e: an addendum. Bull. Amer. Math. So c. 28 (1993), 306–307 9. B. R. Hunt and V. Y. Kal oshin, How pr oje ctions affe ct the dimension sp e ctrum of fr actal me asur es. Nonlinearity 10 (1997), 1031–1046. ALMOST BI-LIPSCHITZ EMBEDDINGS 25 10. B. R. Hunt and V. Y. Kaloshin, R e gularity of emb ed dings of infinite-dimensional fr actal sets into finite-dimensional sp ac es. Nonlinearity 12 (1999), 1263–1275. 11. U. Lang & C. Plaut, Bilipschitz emb e ddings of metric sp ac es int o sp ac e forms. Geom. Dedicata 87 (2001) , 285–307. 12. T. J. Laa kso, Pla ne with A ∞ -weighte d metric not bi -Lipschitz emb ed dable to R N . Bull . London Math. Soc. 3 4 (2002), 667–676. 13. J. Luukk ainen, Assouad dimension: antifr actal metrization, p or ous set s, and homo gene ous me asur es. J. Korean M ath. Soc. 35 (1998), 23–76. 14. R. Ma˜ n´ e, On the dimension of the c omp act invariant sets of c ert ain nonline ar maps. Springer Lecture Notes i n Math., vol. 898, Spri nger, New Y ork, 230–242, 1981. 15. H. Mov ahedi-Lank arani & R. W ells, O n bi-Lipschitz e mb e ddings. Portugaliae Mathematica, to app ear. 16. E.J. Olson, Bouligand dimension and almost Lipschitz emb e ddings. Pacific J. Math. 2 (2002) , 459–474. Dep ar tment of Mathema tics/084, University of Nev ada, Reno, NV 895 57. USA. E-mail addr e ss : ejolson@unr.e du Ma thema tical Institute, University of W ar wick, Coventr y, CV4 7AL. U.K. E-mail addr e ss : jcr@maths.war wick.ac. uk