Substitution Tilings and Separated Nets with Similarities to the Integer Lattice

Substitution Tilings and Separated Nets with Similariti es to the In teger Lattice Y aar Solomon 18.01.2009 Abstract W e show that an y primitive substitution tiling of R 2 creates a sep- arated net which is biLipschitz to Z 2 . Then w e show that if H is a primitive Pisot substitution in R d , for every separ ated net Y , that co r - resp onds to so me tiling τ ∈ X H , there exists a bijection Φ betw een Y and the integer lattice such that sup y ∈ Y k Φ( y ) − y k < ∞ . As a corolla r y we g e t that w e have such a Φ fo r any s e pa rated net that cor- resp onds to a Penrose Tiling . The pro o fs r ely on results of Lacz ko vich, and Burago and Kleiner. 1 In tro duction Definition. A set Y ⊆ R d is called a sep ar ate d net , or a Delone set , if there exist constan ts R, r > 0 such that ev ery ball of radiu s R intersec ts Y and ev ery ball of radius r co ntains at most one p oint of Y . Definition 1.1. Let Y 1 and Y 2 b e separated nets. W e say that a mapping Φ : Y 1 → Y 2 is biLipschitz if there exists a constant C ≥ 1 such that for ev ery y , y ′ ∈ Y 1 w e ha ve 1 C ·   y − y ′   ≤   Φ( y ) − Φ( y ′ )   ≤ C ·   y − y ′   , where k·k is some (any) norm on R d . Φ is called a b ounde d displac ement if sup y ∈ Y 1 k Φ( y ) − y k < ∞ . Consider t wo equ iv alence relat ions on the set of all separate d nets. In one, t w o n ets are equiv alen t if there exists a b iLipsc h itz b ijection b et we en 1 them. In the other, the relatio n holds if th ere is a bijection whic h is also a b ound ed displacement . Since we are dealing with functions b et ween sepa- rated nets, it is easy to v erify that the second rela tion refines the first. A natural question is: Is ev ery sep arated n et in R d biLipsc hitz to Z d ? This question w as fi rst p osed by Gromo v in [Gr93]. It w as answ ered negativ ely in 1998 b y McMullen [McM98] and also ind ep endently b y Burago and Kleiner [BK98]. Their results imply , in particular, that there are separated nets in R d whic h are not a b ounded displacemen t of Z d , not ev en after r escaling. In this pap er we d eal with separated nets w h ic h are obtained from tilings of Euclidean spaces. When a tiling of R d is giv en, b y placing one p oin t in eac h tile, and kee ping the minimal distance prop ert y , one gets a separated net. Since we are studying the equiv alence classes und er b oun ded displace- men t, the p ositions of the p oin ts in the tiles do es not matter. In particular a tiling of R d giv es rise to a separate d net; more precisely , an equiv alence class of nets, in b oth of the ab o ve senses. Our main ob jectiv e is to pro ve the t w o follo wing theorems: Theorem 1.2. Any sep ar ate d net that c orr esp onds to a primitive substitu- tion tiling of R 2 is biLipschitz to Z 2 . Definition 1.3. Let H b e a primitiv e substitution in R d and denote by λ 2 an eigen v alue of A H whic h is s econd in abs olute v alue (see Definitions 2.1, 2.2, 2.5). If | λ 2 | < 1 w e sa y that H is a P isot substitution . Theorem 1.4. L et H b e a Pisot substitution in R d . Then for every substitu- tion tiling of H ther e exists a c onstant β and b ounde d displac ement b etwe en the c orr esp onding sep ar ate d net Y a nd β · Z d . Here we also an s w er a question of Burago and Kleiner ([BK02], p.2). Corollary 1.5. Any sep ar ate d net that is cr e ate d fr om a Penr ose Tiling is a b ounde d displac e ment of β · Z 2 , for some β > 0 (and in p art icular biLi pschitz to Z 2 ). The pro ofs of Theorem 1.2 an d of Theorem 1.4 rely on a result of Burago and Kleiner [BK02] and a r esult of Laczk o vic h [L92] resp ectiv ely . Both of these results d eal w ith the difference b et ween the n u m b er of tiles in a large (b ound ed) set U and the area of U . W e use the Perron F rob enius Theorem and some dynamical prop erties of the matrix of the tiling in order to get go o d estimates for the n um b er of tile s in large sets of certain kind. Then, b y usin g p rop erties of substitution tilings, we fill U with suc h sets, that get smaller and smaller near the b oun dary of U , and get an estimate for the n u m b er of tiles in U . 2 Ac kno w ledgemen t s: This research was su pp orted b y the I srael Science F oundation. This w ork is a p art of th e author’s Master thesis u nder the sup er v ision of Barak W eiss wh ose endless supp ort and guid ance are deeply appreciated. The author also wishes to thank Bruce Kleiner for su ggesting the problem ab out the Penrose Tiling. After [S08] app eared on th e w eb it w as brought to the author’s atten tion that there is another pap er, [DSS 95], where a sk etc h of pr o of for Corollary 1.5 is given. 2 Basic Definitions of T ilings W e use stand ard defin itions of tilings. Similar definitions can b e f ou n d at [GS87], [Ra99], [Ro04]. A set S ⊆ R d is a tile if it is homeomorph ic to a closed d -dimensional ball. A tiling of a set U ⊆ R d is a coun table collection of tiles, with pairwise disjoin t interiors, suc h that their union is equal to U . W e sa y that t wo tiles are tr anslation e quivalent if one is a translation of the other. Represen tativ es of the equiv alence classes are called pr ototiles . A tiling sp ac e , X T , is the s et of all tilings of R d b y prototiles from T . A tiling P of a b ounded set U ⊂ R d is ca lled a p atch . W e call the set U the supp ort of P and we d enote it by supp ( P ). W e extend the equiv alence relation f r om th e last Definition to patc hes and denote by T ∗ the equiv alence class represen tativ es. Substitution Tilings Let ξ > 1 and let T = { S 1 , . . . , S k } b e a set of d -dimensional prototiles. Definition 2.1. A substitution is a m apping H : T → ξ − 1 T ∗ suc h that for ev ery i w e ha v e sup p ( S i ) = supp ( H ( S i )). In other w ords it is a set of dissection rules that sh o ws u s ho w to divide the p rototiles to other pr ototiles from T with a s maller scale. W e extend H to the set of all tiles (in a given tiling), to T ∗ and to an y tiling τ ∈ X T b y applyin g H separately on ev ery tile. Th e constan t ξ is calle d the inflation c onstant of H . Definition 2.2. Let H b e a substitution defined on T . Consider the fol- lo wing set of patc h es: P = { ( ξ H ) m ( T i ) : m ∈ N , i = 1 , . . . , k } . The substitution tiling sp ac e X H is th e set of all tilings of R d that for every patc h P in them there is a p atc h P ′ ∈ P such that P is a sub-patc h of P ′ . Ev ery tiling τ ∈ X H is a substitution tiling of H . 3 Prop osition 2.3. If H is a primitive substitution then X H 6 = ∅ and f or every τ ∈ X H and for every m ∈ N th er e exi sts a tiling τ m ∈ X H that satisfies ( ξ H ) m τ m = τ . Pr o of. See [Ro04]. The construction of substitution tilings is exp lained with more details in [Ro04]. W e denote by H ( − m ) ( τ ) a tiling τ ′ that satisfies ( ξ H ) m τ ′ = τ . Matrices of Substitution Definition. A matrix A is called p ositive , and denoted A > 0, if all its en tries are p ositiv e. A is called nonne gative , an d d enoted A ≥ 0, if the en tries of A are n onnegativ e. A is called primitive if there exists an m ∈ N suc h that A m > 0. Definition 2.4. F or a substitution H , the r epr esentative matrix of H is a k × k matrix B H = ( b ij ), w here b ij is the num b er of p rototiles wh ich are translation equiv alen t to S i in ξ H ( S j ). W e s ay that H is primitive if B H is primitiv e. The matrix B H can b e very large sometimes and do es not describ e ex- actly what w e need here. C on s ider the follo w ing equiv alence relation on prototiles: S i ∼ S j if there exists an isometry O suc h that S i = O ( S j ) and H ( S i ) = O ( H ( S j )) (this is actuall y a condition on the representat iv es, and it is obvio u s that it is we ll d efined). W e call the rep resen tativ es of the equiv alence classes b asic tiles . By this definition, w e can also think of H as a d iss ection rule on the basic tiles and extend it to tiles, p atc hes and tilings as b efore. Definition 2.5. Denote by { T 1 , . . . , T n } , ( n ≤ k ) the set of the b asic tiles. Define the substitution matrix of H to b e an n × n matrix, A H = ( a ij ), where a ij is the num b er of basic tiles in ξ H ( T j ) which are equiv alen t to T i . Example. L et H b e the substitution of the P e nr ose Tiling. Ther e ar e 20 differ ent pr ototiles (r otatio ns and r efle ctions ar e not al lowe d): v v v v v v v v v v           ) ) ) ) ) ) ) ) ) ) H H H H H H H H H H v v v v v v v v v v           ) ) ) ) ) ) ) ) ) ) H H H H H H H H H H       ) ) ) ) ) ) H H H H H H v v v v v v       ) ) ) ) ) ) H H H H H H v v v v v v v v v v v v v v v v       ) ) ) ) ) ) ) ) ) ) H H H H H H v v v v v v           ) ) ) ) ) ) H H H H H H H H H H ) ) ) ) ) )       H H H H H H       v v v v v v H H H H H H ) ) ) ) ) ) v v v v v v 4 Then B H is a 20 × 20 matrix. On the other hand, ther e ar e only two differ ent b asic tiles, with the fol lowing d isse ction rule: ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )                    v v v v v v v v v v v v H H H H H H H H v v v v v v v v v v v v H H H H H H H H H H H H         then A H =  2 1 1 1  , a 2 × 2 matrix. Denote by e i the i ’th elemen t of the standard b asis of R n (or R k ). Then A H ( e i ) is th e i ’th column of A H (and th e same for B H in R k ). T h us, if e i represent s one tile of t yp e i , multiplying the ve ctor e i b y these matrices giv es u s a v ector that repr esents the num b er of basic tiles (pr ototiles) of eac h kind ob tained after applying H on the corresp ond ing tile. By linearit y , this is true f or an y v ector in R n (or R k ). Denote by π : R k → R n the quotient map that defines the relation ∼ . Then by the definition of ∼ , ker( π ) is B H -in v ariant and the follo wing diagram commutes R k π   B H / / R k π   R n A H / / R n Hence, it is easy to v erify th at the eigen v alues of B H , that has an eigenv ector v / ∈ ker( π ), are also eigen v alues of A H . 3 Prop erties of Substitution W e denote b y R n + the set of all nonnegativ e vecto rs in R n . F or a finite set P w e d enote b y # P the num b er of elements of P . W e al so use the notations µ d ( · ) , k ·k ∞ and k·k 2 for d -dimensional Leb esgue measure, the max norm and the Euclidean n orm in R n resp ectiv ely . Definition. Let λ 1 , . . . , λ n b e the eig en v alues of a matrix A . Th e sp e ctr al r adius of A is ρ ( A ) = max i {| λ i |} . F or a nonnegativ e pr imitiv e m atrix A , an eigen v alue λ that satisfies | λ | = ρ ( A ) is called a P err on F r ob enius eige nvalue . 5 Let H b e a p rimitiv e su bstitution with an inflation constant ξ and let τ 0 ∈ X H . Denote b y { T 1 , . . . , T n } th e set of d -dimensional basic tiles. W e denote by λ 1 the Pe rron F rob en ius eigen v alue of A H and let λ 2 b e an eigen- v alue whic h is second in ab s olute v alue. F or ev ery m ∈ N w e denote τ m = H ( − m ) ( τ 0 ), these are substitution tilings with basic tiles { ξ m T 1 , . . . , ξ m T n } . Then for eve ry patc h P of τ m , H m ( P ) is a patc h of τ 0 and supp ( H m ( P )) = supp ( P ). W e denote b y t ( m ) i the num b er of tiles from H m ( P ) which are equiv alen t to T i . Our main ob jectiv e in this section is to p ro ve the follo wing prop osition: Prop osition 3.1. L et H b e a primitive substitution, then ther e ar e c onstants a 1 , a 2 and C 2 , that dep end only on H , such that for every 0 < ǫ < λ 1 − | λ 2 | and for any τ 0 ∈ X H ther e exists an N such that for ev e ry m ≥ N and a p atch P ∈ τ m we have t ( m ) 1 ( a 1 − C 2 δ m ) ≤ # P ≤ t ( m ) 1 ( a 1 + C 2 δ m ) t ( m ) 1 ( a 2 − C 2 δ m ) ≤ µ d ( V ) ≤ t ( m ) 1 ( a 2 + C 2 δ m ) , (1) wher e V = supp ( P ) and (2) δ = | λ 2 | + ǫ λ 1 < 1 . W e start with some notions of matrix theory . S ee also [BP79], [H07]. Prop osition 3.2. L et H b e a primitive substitution with an i nflation c on- stant ξ and let { T 1 , . . . , T n } b e the set of d - dimensional b asic tiles, then (a) ρ ( A H ) i s an eigenvalue of A H , with an algebr aic multipl icity one, and the asso ci ate ei genve ctor is p ositive. (b) If v > 0 is an ei g enve ctor of A H then v c orr esp onds to ρ ( A H ) . (c) Denote by ( v 1 , . . . , v n ) a b asis of gener alize d eigenve ctors of A H with k v i k 2 = 1 for al l i , wher e v 1 is the eige nv e ctor that c orr esp onds to ρ ( A H ) . Then sp { v 2 , . . . , v n } ∩ R n + = { 0 } . (d) In addition, ξ d = ρ ( A H ) , and in p articular ρ ( A H ) > 1 . Pr o of. (a) This is th e well kno w n P err on F rob enius Theorem, also see [Q87], p.91. (b) S ee [BP79] Theorem 2.1.4. 6 (c) This is a w ell kno wn fact, see [H07], 26.1.4.(d). (d) Cons ider th e v ector v =    s 1 . . . s n    where s i is the area of T i . Since the substitution divides eve ry T i to smaller basic tiles, b y u sing Definition 2.5 one can ea sily sho w that ξ d v = A t H v , where A t H is the transp ose matrix of A H . By (b) the pro of is complete. Let A ≥ 0 b e a pr im itive matrix. W e use the n otations λ 1 and λ 2 as b efore. W e al so d enote by v 1 the eigen vect or of λ 1 with k v 1 k 2 = 1 and b y W the generalized eigenspace of the other eigen v alues. F or a v ector u ∈ R n w e write u = β 1 ( u ) v 1 + β 2 ( u ) w u where w u ∈ W with k w u k 2 = 1. Prop osition 3.3. F or a primitive matrix A ≥ 0 ther e exists a c onstant C > 0 , that dep ends only on A , su ch that for every ǫ > 0 ther e exists an N such that for every m ≥ N and a ve ctor u ∈ R n + we have (3)     A m u β 1 ( u ) λ m 1 − v 1     ∞ ≤ δ m , wher e δ as in (2 ). Pr o of. W e denote K = { v ∈ R n + : k v k 2 = 1 } and consider the restrictions of β 1 and β 2 to K . By Prop osition 3.2 (c) we hav e β 1 ( u ) > 0 for ev ery u ∈ K . By the compactness of K w e denote α 1 = min u ∈ K { β 1 ( u ) } > 0 and α 2 = max u ∈ K {| β 2 ( u ) |} > 0 . Denote C ′ = α 2 α 1 , then C ′ > 0 and for ev ery u ∈ R n + w e ha ve: | β 2 ( u ) | β 1 ( u ) = k u k 2 ·    β 2 ( u k u k 2 )    k u k 2 · β 1 ( u k u k 2 ) ≤ α 2 α 1 = C ′ . Notice that there is a constan t a > 0 suc h that for ev ery m ∈ N and w ∈ W with k w k 2 = 1 w e h a ve k A m ( w ) k ∞ ≤ a ( | λ 2 | + ǫ ) m . Th en     A m u β 1 ( u ) λ m 1 − v 1     ∞ =     A m ( β 1 ( u ) v 1 + β 2 ( u ) w u ) − β 1 ( u ) λ m 1 v 1 β 1 ( u ) λ m 1     ∞ =     A m ( β 2 ( u ) w u ) β 1 ( u ) λ m 1     ∞ = | β 2 ( u ) | · k A m ( w u ) k ∞ β 1 ( u ) λ m 1 ≤ C ′ · a ( | λ 2 | + ǫ ) m λ m 1 , whic h completes the pro of for C = C ′ a . 7 Prop osition 3.4. Ther e ar e c onstants C 1 , c 2 , . . . , c n > 0 , that dep end only on H , such that for every 0 < ǫ < λ 1 − | λ 2 | ther e e xists an N 1 such that every m ≥ N 1 satisfies (4)      t ( m ) i t ( m ) 1 − c i      ≤ C 1 δ m for every p atch P of τ m , wher e δ as in (2). Pr o of. By Pr op osition 3.2, λ 1 > 1 and it h as an asso ciated eigen v ector v 1 =      1 c 2 . . . c n      ( c 1 = 1) with c i > 0 for i = 2 , . . . , n . D enote v ′ 1 = v 1 k v 1 k 2 . Fix 0 < ǫ < λ 1 − | λ 2 | , then there is an N suc h that f or ev ery m ≥ N and u ∈ R n + w e ha ve (3) with v ′ 1 instead of v 1 . W e p ick N 1 ≥ N suc h that ev ery m ≥ N 1 satisfies (5) C k v 1 k 2 δ m ≤ 1 2 . F or an arbitrary m ≥ N 1 and a patc h P of τ m , consider the vect or u =    t (0) 1 . . . t (0) n    , w here t (0) i is th e num b er of tiles from P which are equiv alen t to ξ m T i . Obviously u ∈ R n + r { 0 } , then u sati sfies (3) with v ′ 1 instead of v 1 . Hence     k v 1 k 2 · A m H u α 1 ( u ) λ m 1 − v 1     ∞ ≤ C k v 1 k 2 δ m . If we d enote k v 1 k 2 · A m H u α 1 ( u ) λ m 1 =    b ( m ) 1 . . . b ( m ) n    then for i = 2 , . . . , n we ha ve (6)    b ( m ) i − c i    ≤ C k v 1 k 2 δ m and    b ( m ) 1 − 1    ≤ C k v 1 k 2 δ m . In p articular, by (5), 1 2 ≤ b ( m ) 1 ≤ 1 1 2 , f or eve r y m ≥ N 1 . Notice that b y the definitions of u , P and A H w e ha ve (7) A m H u =    t ( m ) 1 . . . t ( m ) n    . 8 Therefore for i = 2 , . . . , n we ha ve      t ( m ) i t ( m ) 1 − c i      ( 7 ) =      b ( m ) i b ( m ) 1 − c i      ≤ 1    b ( m ) 1        b ( m ) i − c i    +    c i − b ( m ) 1 c i     ( 5 ) , ( 6 ) ≤ 2 C k v 1 k 2 (1 + c i ) · δ m ≤ C 1 · δ m , where C 1 = 2 C k v 1 k 2 (1 + max i { c i } ), as requir ed. W e denote by s 1 , . . . , s n the areas of { T 1 , . . . , T n } resp ectiv ely . Define (8) a 1 = n X i =1 c i , a 2 = n X i =1 c i s i and α = a 1 a 2 Pr o of of Pr op osition 3.1. Let 0 < ǫ < λ 1 − | λ 2 | . By Prop osition 3.4 there exists an N = N 1 suc h that (4) holds for ev ery m ≥ N and a patc h P in τ m , for some constan t C 1 . Then f or i = 2 , . . . , n w e ha ve t ( m ) 1 ( c i − C 1 · δ m ) ≤ t ( m ) i ≤ t ( m ) 1 ( c i + C 1 · δ m ) and t ( m ) 1 ( c i s i − C 1 · δ m s i ) ≤ t ( m ) i s i ≤ t ( m ) 1 ( c i s i + C 1 · δ m s i ) . Therefore t ( m ) 1 n X i =1 c i − nC 1 · δ m ! ≤ n X i =1 t ( m ) i ≤ t ( m ) 1 n X i =1 c i + nC 1 · δ m ! t ( m ) 1 n X i =1 c i s i − C 1 · δ m n X i =1 s i ! ≤ n X i =1 t ( m ) i s i ≤ t ( m ) 1 n X i =1 c i s i + C 1 · δ m n X i =1 s i ! . Th us, according to (8), for C 2 = max { C 1 n, C 1 P n i =1 s i } we get (1) as re- quired. 4 The Main Results W e pro ve Theorem 1.2 by sho wing that substitution tilings create s eparated nets that satisfy th e conditions of th e follo win g theorem: 9 Theorem 4.1 (Burago and Kleiner [BK02]) . L et Y b e a sep ar ate d net in R 2 . F or a r e al numb er α > 0 and a sq u ar e B with inte ger c o or dinates define: e α ( B ) = max  α · µ 2 ( B ) #( B ∩ Y ) , #( B ∩ Y ) α · µ 2 ( B )  E α (2 i ) = sup  e α ( B ) : B as ab ove with an e dge of leng th 2 i  . If ther e exists an α > 0 such that the pr o duct Q ∞ j =1 E α (2 j ) c onver ges, then Y is biLipschitz to Z 2 . Pr o of of The or em 1.2. It suffices to sho w that there are constants C 1 , k 1 > 0 and ω < 1 su c h that for ev ery square B with an edge of length 2 j = k ≥ k 1 w e ha ve (9) | α · µ 2 ( B ) − #( B ∩ Y ) | #( B ∩ Y ) ≤ C 1 · ω j and | α · µ 2 ( B ) − #( B ∩ Y ) | α · µ 2 ( B ) ≤ C 1 · ω j . Then we get, for all large enough j ,   E α (2 j ) − 1   ≤ C 1 · ω j , whic h implies the con v ergence of the p ro du ct. By Prop osition 3.2 we can pic k an ǫ > 0 such th at λ 1 > | λ 2 | + ǫ . By Prop osition 3.1 there is an N 1 suc h that for ev ery m ≥ N 1 and a p atc h P in τ m , (1) holds. Let N ≥ N 1 suc h that for ev er y m ≥ N we h a ve (10) C 2 · δ m ≤ 1 2 min { a 1 , a 2 } ( a 1 , a 2 as in (8) , δ as in (2)) . W e p ick k ′ = ξ 2 N . Let B b e an arbitrary square in R 2 with an edge of length k ≥ k ′ . Let m ∈ N suc h that ξ 2 m ≤ k < ξ 2 m +2 , then m ≥ N . Consider the patc h P = { T ∈ τ m : T ⊆ B } , then P satisfies (1), wher e V = s upp ( P ) as b efore. Let R and r b e constan ts suc h that ev ery b all of d iameter R con tains a tile of τ 0 and ev ery tile of τ 0 con tains a cub e of area r . F r om the definition of P , for ev ery x ∈ B th at satisfies d ( x, ∂ B ) ≥ R · ξ m , the tile of τ m that co v ers x m u st b e in P . Th en V con tains a square with an edge of length k − 2 R · ξ m . Since R · ξ m ≤ R · √ k , V con tains a square with an edge of length k − 2 R · √ k . If so, there is a k ′′ suc h that for ev ery k ≥ k ′′ w e h a ve 10 µ 2 ( V ) ≥ 1 2 k 2 . Then, by (1), th ere is a co n stan t b 1 > 0 su ch that for every k ≥ k ′′ w e ha ve (11) b 1 · k 2 ≤ t ( m ) 1 . Define k 1 = max { k ′ , k ′′ } . Consider squares B with an edge of length k ≥ k 1 . W e w ant to estimate #( B ∩ Y ) and µ 2 ( B ). Define the follo wing patc h of τ 0 : P 1 = { T ∈ τ 0 : T ∩ B 6 = ∅} V 1 = sup p ( P 1 ) . A similar explanation to the one ab o ve give s the estimate V 1 r V ⊆ { x : d ( x, ∂ B ) ≤ R · ξ m } ⊆ { x : d ( x, ∂ B ) ≤ R · √ k } . Then µ 2 ( V 1 r V ) ≤ 4 R · k √ k and so #(( V 1 r V ) ∩ Y ) ≤ 4 R · k √ k r . Th erefore # P ≤ #( B ∩ Y ) ≤ # P + #(( V 1 r V ) ∩ Y ) , µ 2 ( V ) ≤ µ 2 ( B ) ≤ µ 2 ( V ) + µ 2 ( V 1 r V ) . Hence, b y (1) t ( m ) 1 ( a 1 − C 2 δ m ) ≤ #( B ∩ Y ) ≤ t ( m ) 1 ( a 1 + C 2 cδ m ) + 4 R · k √ k r , t ( m ) 1 ( a 2 − C 2 δ m ) ≤ µ 2 ( B ) ≤ t ( m ) 1 ( a 2 + C 2 δ m ) + 4 R · k √ k . Therefore, for α as in (8) we h a ve α · µ 2 ( B ) − #( B ∩ Y ) #( B ∩ Y ) ≤ α ( t ( m ) 1 ( a 2 + C 2 δ m ) + 4 R · k √ k ) − t ( m ) 1 ( a 1 − C 2 δ m ) t ( m ) 1 ( a 1 − C 2 δ m ) = t ( m ) 1 C 2 δ m ( α + 1) + 4 αR · k √ k t ( m ) 1 ( a 1 − C 2 δ m ) ( 11 ) ≤ δ m · C 2 ( α + 1) a 1 − C 2 δ m + 1 √ k · 4 αR b 1 ( a 1 − C 2 δ m ) . In the same w ay we get s im ilar inequalities for α · µ 2 ( B ) − #( B ∩ Y ) #( B ∩ Y ) and then for | α · µ 2 ( B ) − #( B ∩ Y ) | α · µ 2 ( B ) . Hence, consid ering (10), ther e is a constan t C ′ 1 suc h that max  | α · µ 2 ( B ) − #( B ∩ Y ) | α · µ 2 ( B ) , | α · µ 2 ( B ) − #( B ∩ Y ) | #( B ∩ Y )  ≤  δ m + 1 √ k  · C ′ 1 . Notice that m w as chosen in a wa y that ξ 2 m ≤ k , thus ξ m ≤ √ k . Sin ce we are lo oking on s quares with k = 2 j , w e get that m ≤ j · log ξ √ 2. Th er efore, ω = max { δ log ξ √ 2 , 1 / √ 2 } satisfies the condition in (9), with C 1 = 2 C ′ 1 . 11 W e now turn to the pr o of of Theorem 1.4. Th e pro of relies on th e follo wing theorem: Theorem 4.2 (Laczk o vic h [L92]) . F or a sep ar ate d net Y ⊆ R d and α > 0 the fol lowing statements ar e e quivalent: (i) Ther e is a p ositive c onstant C such that for every finite union of unit cub es U we have (12) | #( Y ∩ U ) − αµ d ( U ) | ≤ C · µ d − 1 ( ∂ U ) . (ii) Ther e is a b ounde d displac ement φ : Y → α − 1 /d Z d . F or the pro of of Theorem 1.4 w e will n eed the tw o follo wing lemmas: Lemma 4.3. Ther e i s a c onstant C 3 such that for every 0 < ǫ < λ 1 − | λ 2 | ther e exists an N such that for every m ≥ N and a tile T in τ m we have (13) | #( T ∩ Y ) − αµ d ( T ) | ≤ C 3 · ( λ 2 + ǫ ) m , wher e α a s in (8). Pr o of. W e th in k of T as a patc h in τ m and denote P 0 = H m ( T ), the patc h in τ 0 with supp ( P 0 ) = T . Then b y Prop osition 3.1 we h a ve (1 ) with T in stead of P , for ev ery m ≥ N 1 . O n th e other hand, T is equiv alen t to ξ m T i for some i ∈ { 1 , . . . , n } , then µ d ( T ) = ( ξ d ) m · s i . Th en t ( m ) 1 ( a 2 − C 2 δ m ) ≤ ( ξ d ) m · s i , whic h implies, for eve r y m whic h is greater than some N 2 , (14) t ( m ) 1 ≤ ( ξ d ) m · s i a 2 − C 2 δ m ≤ C ′ 3 ( ξ d ) m ( 3.2 )( c ) = C ′ 3 λ m 1 . According to (1) w e ha v e #( T ∩ Y ) − αµ d ( T ) ( 1 ) ≤ t ( m ) 1 ( a 1 + C 2 δ m ) − αt ( m ) 1 ( a 2 − C 2 δ m ) ( 8 ) = t ( m ) 1 C 2 δ m (1 + α ) , and in a similar w a y we get it for αµ d ( T ) − #( T ∩ Y ). Then | #( T ∩ Y ) − αµ d ( T ) | ≤ t ( m ) 1 C 2 δ m (1 + α ) ( 14 ) ≤ C ′ 3 λ m 1  | λ 2 | + ǫ λ 1  m C 2 (1 + α ) . All this is tru e for ev ery m ≥ N , where N = max { N 1 , N 2 } . Then for C 3 = C ′ 3 · C 2 (1 + α ) we get the required inequ alit y . 12 Lemma 4.4. Ther e i s a c onstant C , that dep e nds only on the dimension d , such that for any s > 1 (15) µ d ( { x ∈ U : d ( x, ∂ U ) ≤ s } ) ≤ C · s d · µ d − 1 ( ∂ U ) holds for any finite union of d -dimensional cub es U . Pr o of. Th is a d irect result of Lemma 2.1 and Lemma 2.2 of [L92]. Pr o of of The or em 1.4. First we claim that it is s ufficien t to show inequalit y (12) for ev ery set U wh ic h is a finite union of cub es with an edge of length k , for some constan t k ∈ N . Then indeed, b y rescaling the wh ole picture b y a factor of 1 k , w e get (12) for the net 1 k · Y with 1 k · U instead of U , α k d instead of α and a differen t constant C 1 . Since 1 k · U is a fi nite un ion of unit cub es, by [L92], we get a b ound ed displacemen t Φ ′ : 1 k · Y → α 1 /d 1 k · Z d , whic h implies the existence of th e required Φ. W e pick the constan t α as in (8 ). Since H is a Pisot substitution, w e fix ǫ > 0 suc h that | λ 2 | + ǫ < 1 (By Prop osition 3.2 λ 1 = ξ d > 1). Then let N b e such that (13) h olds for every tile T ∈ τ m where m ≥ N . Let R and r b e constan ts suc h th at ev ery b all of diameter R con tains a tile of τ 0 and ev ery tile of τ 0 con tains a cub e of area r . Let U b e a finite union of cub es in R d with an edge of length k =  R · ξ N  . Let m b e the maximal in teger su c h that U con tains a tile of τ m . Then b y the defin ition of k w e ha v e m ≥ N . Define the follo w in g s equ ence of patc h es: P m = { T ∈ τ m : T ⊆ U } and for decreasing l = m − 1 , . . . , N P l =    T ∈ τ l : int ( T ) ⊆ U r m [ j = l +1 V j    , where V l = sup p ( P l ) and int ( T ) is the in terior of T . Define V ∂ = U r S m j = N V j , then w e get a partition of U to la yers that in tersect only at their b oundaries: (16) U = m [ l = N V l ! ∪ V ∂ , whic h implies µ d ( U ) = m X l = N µ d ( V l ) ! + µ d ( V ∂ ) . 13 W e no w estimate # P l . Notice that for every x ∈ U , if d ( x, ∂ U ) ≥ R · ξ l then any ball of diameter R · ξ l that contai ns x is con tained in U . Then the til e of τ l that conta ins x is con tained in U . In particular we get it for l = m + 1. But sin ce no tile of τ m +1 is con tained in U , w e d educe that d ( x, ∂ U ) < R · ξ m +1 for eve ry x ∈ U . Th er efore µ d ( U ) ≤ µ d  x ∈ U : d ( x, ∂ U ) < R · ξ m +1  ( 15 ) ≤ C · ( Rξ m +1 ) d µ d − 1 ( ∂ U ) . Since eve ry tile of τ m con tain a cub e of area r ( ξ d ) m w e ha ve # P m ≤ C ( Rξ m +1 ) d µ d − 1 ( ∂ U ) r ( ξ d ) m = C ( Rξ ) d µ d − 1 ( ∂ U ) r . In a similar wa y , for ev ery l , if d ( x, ∂ U ) ≥ R · ξ l +1 then the tile of τ l +1 that co vers x is in P l +1 , thus x / ∈ V l . Hence for eve ry x ∈ V l w e ha ve d ( x, ∂ U ) < R · ξ l +1 and so µ d ( V l ) ≤ µ d n x ∈ U : d ( x, ∂ U ) < R · ξ l +1 o ( 15 ) ≤ C · ( Rξ l +1 ) d µ d − 1 ( ∂ U ) , whic h implies (17) # P l ≤ C ( Rξ l +1 ) d µ d − 1 ( ∂ U ) r ( ξ d ) l = C ( Rξ ) d µ d − 1 ( ∂ U ) r . Similarly we get µ d ( V ∂ ) ≤ C ( Rξ N ) d µ d − 1 ( ∂ U ) #( V ∂ ∩ Y ) ≤ C ( Rξ N ) d µ d − 1 ( ∂ U ) r . (18) If we d enote b y T ( l ) tiles of τ l , then for ev ery l ≥ N we h a ve | #( V l ∩ Y ) − αµ d ( V l ) | =       X T ( l ) ⊆ V l #( T ( l ) ∩ Y ) − α · X T ( l ) ⊆ V l µ d ( T ( l ) )       ≤ X T ( l ) ∈ P l    #( T ( l ) ∩ Y ) − αµ d ( T ( l ) )    ( 13 ) , ( 17 ) ≤ C ( Rξ ) d µ d − 1 ( ∂ U ) r · C 3 · ( | λ 2 | + ǫ ) l ≤ C 4 · ( | λ 2 | + ǫ ) l · µ d − 1 ( ∂ U ) , 14 where C 4 = C · C 3 · ( Rξ ) d r . Therefore, according to (18), we den ote C 5 = max n α · C ( Rξ N ) d , ( Rξ N ) d r o and get | #( U ∩ Y ) − αµ d ( U ) | ( 16 ) ≤ " m X l = N | #( V l ∩ Y ) − αµ d ( V l ) | # + | #( V ∂ ∩ Y ) − αµ d ( V ∂ ) | ≤ " m X l = N C 4 · ( | λ 2 | + ǫ ) l · µ d − 1 ( ∂ U ) # + C 5 · µ d − 1 ( ∂ U ) ≤ C 1 · µ d − 1 ( ∂ U ) , where C 1 = C 4  P ∞ l =1 ( | λ 2 | + ǫ ) l  + C 5 . Pr o of of Cor ol lary 1.5. If we denote by H the subs titution of the P enrose Tiling then A H =  2 1 1 1  . In th is case w e ha v e λ 2 = 3 − √ 5 2 < 1. By T h eorem 1.4 the pr o of is complete. References [BP79] A. Berman and R. J. Plemmons, Nonnegati v e matrices in the mathematical sciences, Academic Press, New Y ork, 1979 . [BK98] D. Burago and B. Kleiner, Sep ar ate d nets in Euclide an sp ac e and Jac obians of biLipschitz map , Geom. F unc. Anal. 8 (1998), no.2, 273-2 82. [BK02] D. Burago and B. Kleiner, R e c tifying sep ar at e d nets , Geom. F unc. Anal. V ol.12 (200 2) 80-92. [DSS95] W. A. Deub er, M. Simonovits and V. T . Sos, A note on p ar adoxic al metric sp ac es , S tudia Sci.Hung.Math. 30 (1995), 17–2 3. [Gr93] M. Gromo v, Asymp totic inv ariants of infin ite groups , Geometric Group Theory V ol.II (G. Niblo and M. Roller eds.), London Math. So c. Lecture Notes , 182, Cambridge Univ. Press (1993). [GS87] Brank o Gru b aum and G.C. Shephard , Tilings and patterns , W.H.F reeman and Company , New Y ork, 1987. [H07] L. Hogb en (Ed.), Handb o ok of linear algebra, Ch apman and Hall/CR C Press, 2007. MR2279160 (2007j:15001 ). 15 [L92] M. Laczk ovic h, Uni f ormly spr e ad discr ete sets in R d , J. London Math. So c. (2) 46 (1992 ) 39-57. [McM98] C. T. McMullen, Lipschitz maps and nets in Euclide an sp ac e , Geom. F unc. An al. 8 (1998) , no.2, 273-282 . [Q87] M. Queffelec, Substitution dynamic al systems-Sp e ctr al analysis , Lecture n otes in m athematics, vo l.1294 , Sp ringer-V erlag, Berlin, 1987. [Ra99] C. Radin, Miles of tiles , Amer. Math. So c., Pro vidence, RI (1999). [Ro04] E. A. Robinson, Jr. Symb olic dynamics and tilings of R n , Pro c. Symp os. App l. Math. V ol.60 (2004), 81-11 9. [S08] Y. S olomon, The net cr e ate d fr om the Penr ose Tiling is biLipschitz to the inte ger lattic e , arXiv:0711 .3707 v1 (2008). 16