An inverse problem in number theory and geometric group theory

AN INVERSE PR OBLEM IN NUMBER THEOR Y AND GEOMETRIC GR OUP THEOR Y MEL VYN B. NA THA NSON Abstract. This paper descri bes a new link betw een com binatorial num ber theory and geometry . The main result state s that A is a finite set of r elativ ely prime positive integ ers if and only i f A = ( K − K ) ∩ N , where K is a compact set of real n um b ers suc h that for every x ∈ R the re exists y ∈ K wi th x ≡ y (mod 1). In one direction, give n a finite set A of relatively prime posi tiv e int egers, the proof constructs an appropriate compac t set K suc h that A = ( K − K ) ∩ N . In the other direction, a strong form of a fundamen tal r esult in geometric group theory is applied to prov e that ( K − K ) ∩ N is a finite set of r el atively prime positive inte gers if K satisfies the appropriate geometrical conditions. Some r elated results and open problems are also di s cussed. 1. From comp act sets to integers The o b ject of this note is to describ e a new connec tio n b etw een n um b er theo ry and geometr y . Let R , Z , and N deno te the real num b ers, integers, a nd p ositive int egers, resp ectively . F or every x ∈ R , let [ x ] ∈ Z and ( x ) ∈ [0 , 1) denote the int eger part and fractional part of x . Let Z n denote the additive gro up of n - dimensional lattice p oints in the Euclidean space R n . W e recall the following definitions. The set A of integers is r elatively prime , denoted gcd( A ) = 1, if A is nonempt y and the elemen ts of A have no commo n factor greater than 1. E quiv alently , A is r elatively prime if A generates the additiv e group Z . The set A of n -dimensiona l lattice p oints is re la tively prime if the elements of A genera te the additive g roup Z n . Let H b e a subg r oup of a m ultiplica tive gro up G , and let x a nd y b e elements of G . W e s ay that x a nd y are c ongruent mo dulo H , denoted x ≡ y (mo d H ), if xy − 1 ∈ H . If the group G is additive, then x ≡ y (mo d H ) if x − y ∈ H . F or example, let G = R and H = Z . The real n umbers x and y ar e congruent modulo Z , that is, x ≡ y (mo d Z ) or , in more traditional notatio n, x ≡ y (mo d 1 ), if and only if they ha ve the sa me fractional par t. In a multip licative g roup G with iden tity e , the differ enc e set of a nonempt y subset K of G is K K − 1 = { xy − 1 : x, y ∈ K } . Date : No ve mber 12, 2018. 2000 Mathematics Subje ct Classific ati on. Primar y 11A05, 11B75, 11P21, 20F65. Key wor ds and phr ases. Relativ ely prime i n tegers, combinatorial num ber theory , additiv e num- ber theory , geometric group theory . Supported i n part by a gran t from the PSC-CUNY Research Aw ard Program. This paper wa s written while the author w as a visiting fello w in the mathematics departmen t at Princeton Unive rsity . 1 2 MEL VYN B. NA THANSON In an additive ab elian group G , the differ enc e set of a s ubset K o f G is K − K = { x − y : x, y ∈ K } . Note that e ∈ K K − 1 and that the difference se t is s ymmetric: z ∈ K K − 1 if and only if z − 1 ∈ K K − 1 (resp. z ∈ K − K if and only if − z ∈ K − K ). W e are interested in sets of integers co ntained in difference sets of sets of rea l nu mbers. O ur main theorem gives a ge o metric condition for a finite set of p ositive int egers to b e r elatively prime. The geo metry uses the c o ncept of an N - set, which is a compact subse t K of R n such that for every x ∈ R n there exists y ∈ K with x ≡ y (mo d Z n ). Theorem 1. L et A b e a finite set of p ositive int e gers. The s et A is r elatively prime if and only if ther e exists an N -set K in R su ch that A = ( K − K ) ∩ N . In Section 2, w e solve the inv erse problem: Given a finite set of relatively prime int egers, we construct an N -set K in R such that A = ( K − K ) ∩ N . In Section 3 we prov e that if K is an N -s et in R , then A = ( K − K ) ∩ N is a finite set of relatively prime p ositive integers. The pro of uses the “fundamental observ ation of geometric group theo ry” (Theorem 5), which is reviewed in Section A. Ideas from geometric group theory ha ve b e en used recently to obtain new results in n umber theory (e.g. Natha nson [4, 5, 6]), and should contin ue to be use ful. The b o ok of de la Harp e, T opics in Ge ometric Gr oup The ory [1], is an excellent survey o f this sub ject. Theorem 5 was discovered and prov ed independently by Efremovi ˇ c [2], ˇ Sv arc [7], a nd Milnor [3]. 2. The inverse problem In this section we pr ov e that ev ery finite set of relatively prime po s itive integers can be realized as the difference set o f an N -set. The construction dep ends on the following simple observ ation. Lemma 1. L et K b e a set of r e al numb ers, and let a ∈ Z \ { 0 } . Then a ∈ K − K if and only if ther e is a two-element subset { x, y } of K such that ( x ) = ( y ) and a = [ x ] − [ y ] . Pr o of. F or any no n-zero in teger a , we hav e a ∈ K − K if a nd only if there exist x, y ∈ K such that x 6 = y and a = x − y = [ x ] − [ y ] + ( x ) − ( y ) . Since [ x ] − [ y ] ∈ Z and − 1 < ( x ) − ( y ) < 1, it follows that ( x ) − ( y ) = 0 and a = [ x ] − [ y ]. The se t { x, y } satisfies the conditions of the Lemma.  Here are three examples. W e asso ciate the set A 1 = { 2 , 5 } with the N -set K ( A 1 ) = [0 , 1 / 3] ∪ [2 + 1 / 3 , 2 + 2 / 3] ∪ [4 + 2 / 3 , 5] . There ar e three tw o -element subsets { x, y } of K ( A 1 ) such that x and y hav e the same fractiona l part: { 1 / 3 , 2 + 1 / 3 } , { 2 + 2 / 3 , 4 + 2 / 3 } , and { 0 , 5 } . The set A 2 = { 6 , 10 , 15 } a rises from the N - set K ( A 2 ) = [0 , 1 / 3] ∪ [15 + 1 / 3 , 15 + 2 / 3] ∪ [9 + 2 / 3 , 10] . The co mplete list o f the tw o- element subsets { x, y } of K ( A 2 ) suc h that x a nd y hav e the sa me fractional part is: { 1 / 3 , 15 + 1 / 3 } , { 15 + 2 / 3 , 9 + 2 / 3 } , and { 0 , 10 } . NUMBER THEOR Y AND GEOMETRIC GR OUP THEOR Y 3 F or the set A 3 = { 18 , 28 , 63 } , the N -set K ( A 3 ) = 9 [ i =0 [ − 18 i + i/ 13 , − 18 i + ( i + 1) / 13 ] ∪ [ − 99 + 10 / 1 3 , − 99 + 11 / 13] ∪ [ − 3 6 + 11 / 1 3 , − 36 + 12 / 13 ] ∪ [27 + 12 / 13 , 28] satisfies ( K ( A 3 ) − K ( A 3 )) ∩ N = A 3 . There a re exa ctly 12 tw o -element subsets { x, y } of K ( A 3 ) suc h that x and y hav e the same fra ctional part. In the following Lemma we co nstruct an imp ortant example of an N - set on the real line, and its asso ciated difference set of in tegers. Lemma 2. F or the p ositive inte ger w , let λ 0 < λ 1 < · · · < λ w − 1 < λ w b e a strictly incr e asing se quenc e of r e al numb ers such that λ w = λ 0 + 1 and let b 0 , b 1 , . . . , b w − 1 b e a se quenc e of inte gers such that b k − 1 6 = b k for k = 1 , . . . , w − 1 and 1 + b w − 1 6 = b 0 . The s et K ′ = w − 1 [ k =0 [ b k + λ k , b k + λ k +1 ] is an N -set, and ( K ′ − K ′ ) ∩ N = {| b k − b k − 1 | : k = 1 , . . . , w − 1 } ∪ {| 1 + b w − 1 − b 0 |} is a fi n ite set of r elatively prime p ositive inte gers. Pr o of. The set K ′ is co mpact b ecause it is a finite union of closed interv als , and an N -set b ecause w − 1 [ k =0 [ λ k , λ k +1 ] = [ λ 0 , λ w ] = [ λ 0 , λ 0 + 1] . Let A b e the finite set o f p ositive in tegers contained in the difference set K ′ − K ′ . Since {{ b k − 1 + λ k , b k + λ k } : k = 1 , . . . , w − 1 } ∪ {{ b 0 + λ 0 , b w − 1 + λ w }} is the set of all tw o- element subsets { x, y } of K ′ with ( x ) = ( y ), it follo ws that A = ( K ′ − K ′ ) ∩ N = {| b k − b k − 1 | : k = 1 , . . . , w − 1 } ∪ {| 1 + b w − 1 − b 0 |} . Cho ose ε k ∈ { 1 , − 1 } suc h that | b k − b k − 1 | = ε k ( b k − b k − 1 ) for k = 1 , . . . , w − 1, and ε w ∈ { 1 , − 1 } s uch that | 1 + b w − 1 − b 0 | = ε w (1 + b w − 1 − b 0 ) . 4 MEL VYN B. NA THANSON Since 1 = ε w | 1 + b w − 1 − b 0 | − w − 1 X k =1 ε k | b k − b k − 1 | it follows that A is a finite set of relatively prime p ositive in tegers.  Theorem 2. If A is a finite set of r elatively prime p ositive int e gers, then ther e is an N -set K s u ch that A = ( K − K ) ∩ N . Pr o of. Since the element s of A are relatively pr ime, we ca n write 1 as an integral linear combination o f elements of A . Th us, there exist pair wise distinct integers a 1 , . . . , a h in A , p ositive int egers w 1 , . . . , w h , and ε 1 , . . . , ε h ∈ { 1 , − 1 } suc h that (1) h X i =1 ε i w i a i = 1 . Rewriting (1), w e obtain (2) ε h a h = 1 + h − 1 X i =1 w i ( − ε i a i ) + ( w h − 1)( − ε h a h ) . Let w 0 = 0 and w = P h i =1 w i . F or j = 1 , 2 , . . . , w , we define integers ˜ a j as follows: If w 1 + · · · + w i − 1 + 1 ≤ j ≤ w 1 + · · · + w i − 1 + w i then ˜ a j = − ε i a i . It follows that 1 + w X j =1 ˜ a j = 1 + h X i =1 w i ( − ε i a i ) = 0 . F or k = 0 , 1 , . . . , w , we consider the integers (3) b k = k X j =1 ˜ a j and real num ber s λ k = k w . Then b 0 = 0, 0 = λ 0 < λ 1 < · · · < λ w = 1 and, for k = 1 , . . . , w , b k − b k − 1 = ˜ a k 6 = 0 . It follows from (1) and (3) that 1 + b w − 1 = 1 + w − 1 X j =1 ˜ a j = 1 + w X j =1 ˜ a j − ˜ a w = − ˜ a w = ε h a h 6 = 0 = b 0 . Construct the N - set K ′ = w − 1 [ k =0 [ b k + λ k , b k + λ k +1 ] . NUMBER THEOR Y AND GEOMETRIC GR OUP THEOR Y 5 By Lemma 2, ( K ′ − K ′ ) ∩ N = {| b k − b k − 1 | : k = 1 , . . . , w − 1 } ∪ {| 1 + b w − 1 − b 0 |} = {| ˜ a k | : k = 1 , . . . , w − 1 } ∪ {| − ˜ a w |} = { a i : i = 1 , . . . , h } . Let card( A ) = ℓ. If ℓ = h , then A = { a 1 , . . . , a h } and we set K = K ′ . Suppo se that ℓ > h and A \ { a 1 , . . . , a h } = { a h +1 , . . . , a ℓ } 6 = ∅ . Since i w ( ℓ − h + 1) ∈  0 , 1 w  = [ b 0 + λ 0 , b 0 + λ 1 ] ⊆ K ′ for i = 1 , 2 , . . . , ℓ − h , it follows tha t K = K ′ ∪  a h + i + i w ( ℓ − h + 1) : i = 1 , 2 , . . . , ℓ − h  is an N -set suc h that A = ( K − K ) ∩ N . This completes the pro of.  Let A b e a finite set of relatively prime pos itive in teg ers. W e define the weight of a repres e n tation of 1 in the form (1) by h X i =1 w i + car d( A ) − h. W e define the additive weight of A , denoted Add( A ) as the sma lle s t weigh t o f a representation of 1 b y elements o f A . Note that Add( A ) ≥ card( A ) for a ll A , and Add( A ) = card( A ) if a nd only if there exist distinct in tegers a 1 , . . . , a h ∈ A and ε 1 , . . . , ε h ∈ { 1 , − 1 } s uch that P h i =1 ε i a i = 1. W e define the w e ig ht of an N -set K as the num b er o f connected comp onents of K , and the ge ometric weight o f A , denoted Geo( A ) as the smallest weigh t of an N -set K such that A = ( K − K ) ∩ N . The following result follows immediately from the pro of of Theorem 2. Corollary 1. L et A b e a finite set of r elatively prime p ositive inte gers. Then Geo( A ) ≤ Add( A ) . There exist sets A s uch that Geo( A ) < Add( A ) . F or example, if A = { 1 , 2 , 3 , . . . , n } , then K = [0 , n ] is an N -set of weigh t 1 such that ( K − K ) ∩ N = A , and so Geo( A ) = 1 < n = Add( A ). 3. Rela tivel y prime sets of la ttice points In this s ection we o btain the con verse of Theo rem 2. Theorem 3. If K is an N -set in R , then A = ( K − K ) ∩ N is a fin ite set of r elatively prime p ositive inte gers. W e prove this result in n dimensions . Theorem 4. If K is an N - set in R n , then A = ( K − K ) ∩ Z n is a fi nite set of r elatively prime lattic e p oints. Note the nec e ssity of the c o mpactness co ndition. F or n ≥ 1, the non-co mpact set K = [0 , 1) n has the prop erty that for all x ∈ R n there exists y ∈ K with x ≡ y (mo d Z n ), but ( K − K ) ∩ Z n = { 0 } . 6 MEL VYN B. NA THANSON Pr o of. The pro of uses a result called “the fundamental obs erv ation of geometr ic group theory” (Theorem 5 ). W e discuss this in App endix A. The additive gr oup Z n acts isometr ic ally and prop erly disc o ntin uously on R n by translation: ( g , x ) 7→ g + x for g ∈ Z n and x ∈ R n . The quotient space Z n \ R n is the n -dimensional torus, which is compact, and so the group action Z n y R n is co-compact. Let π : R n → Z n \ R n be the quotient map. Then π ( x ) = h x i is the orbit of x for all x ∈ R n . If K is an N -set in R n , then K is compact, and for ev ery x ∈ R n there exists y ∈ K such that x ≡ y (mo d Z n ). This means that π ( y ) = h x i , and so π ( K ) = Z n \ R n . Applying Theorem 5 to the set K , we conclude that the set A = { a ∈ Z n : K ∩ ( a + K ) 6 = ∅} is a finite set of generators for Z n . Mor eov er, a ∈ A if a nd only if a ∈ Z n and ther e exists x ∈ K such that x ∈ a + K , that is, x = a + y for some y ∈ K . Equiv a lent ly , a ∈ A if and only if a = x − y ∈ ( K − K ) ∩ Z n . This prov es Theorem 4.  The symmetry o f the difference set immediately implies Theor em 3. W e can state the fo llowing gener al in verse problem in geometric gr oup theory: If A is a finite set of generator s for a group G suc h that A is symmetric and cont ains the identit y o f G , do es there exis t a geometric actio n of G on a metric space X with quo tient map π : X → G \ X such that A = { a ∈ G : K ∩ aK 6 = ∅} for some compact set K with π ( K ) = G \ X ? If X is a group and G is a subgroup of X that acts on X b y left translation, then { a ∈ G : K ∩ aK 6 = ∅ } = K K − 1 ∩ G. If G = Z n and X = R n , then the inv erse problem is to determine if ev ery finite symmetric set relatively prime la ttice p o ints that contains 0 is of the fo rm ( K − K ) ∩ Z n for some N -set K . In this pap er we pro ved that the ans wer is “yes” for G = Z , but the answer is not known for higher dimens io n. In pa r ticular, the inv erse problem for lattice p oints is op en for n = 2. W e would like a descr iption of the sets of lattice p oints tha t can b e represented in the form ( K − K ) ∩ Z 2 for some N -set K . Appendix A. The fundament al obser v a tion of geometric group theor y The pro of of Theorem 4 is a n applica tion o f what is often called the “fundamental observ a tion of g eometric gro up theo ry” [1, Chapter IV, pp. 87 – 88]. W e shall describ e this r esult, which is not well kno wn to num b er theorists. W e b eg in b y intro ducing the clas s of b o undedly compact g eo desic metric spaces. The Heine-Borel theorem states that, in E uclidean space R n with the usual metric, a closed and b ounded s et is compact. W e s ha ll call a metric s pace ( X , d ) b oun de d ly c omp act if ev ery closed and bounded subset of X is compact. Eq uiv alently , X is bo undedly compact if ev ery closed ball B ∗ ( x 0 , r ) = { x ∈ X : d ( x 0 , x ) ≤ r } is compact for all x 0 ∈ X and r ≥ 0. Boundedly compact metric spaces are a lso called pr op er metric s paces. A metric space ( X , d ) is ge o desic if, for all po ints x 0 , x 1 ∈ X with x 0 6 = x 1 , there is an isometry γ fro m an in terv al [ a, b ] ⊆ R in to X such that γ ( a ) = x 0 and γ ( b ) = x 1 . Thu s, if t, t ′ ∈ [ a, b ], then d ( γ ( t ) , γ ( t ′ )) = | t − t ′ | . In par ticular, NUMBER THEOR Y AND GEOMETRIC GR OUP THEOR Y 7 d ( x 0 , x 1 ) = d ( γ ( a ) , γ ( b )) = b − a . F o r example, let x 0 , x 1 ∈ R n with | x 1 − x 0 | = T . Define γ : [0 , T ] → R n by γ ( t ) = x 0 + t T ( x 1 − x 0 ) . Then γ (0) = x 0 , γ ( T ) = x 1 , and | γ ( t ) − γ ( t ′ ) | =      x 0 + t T ( x 1 − x 0 )  −  x 0 + t ′ T ( x 1 − x 0 )      =      t − t ′ T  ( x 1 − x 0 )     = | t − t ′ | . Thu s, R n is a b oundedly compact geo desic metric s pace. Let G b e a g roup that acts on a metric space ( X , d ). W e say that the gro up G acts isometric al ly on X if the function x 7→ g x is an iso metry on X for every g ∈ G . The group action is called pr op erly disc ont inuous if, for every compa ct subset K of X , there are only finitely man y a ∈ G such that K ∩ aK 6 = ∅ . Let A = { a ∈ G : K ∩ aK 6 = ∅ } . Then A 6 = ∅ b eca us e e ∈ A . Since K ∩ a − 1 K = a − 1 ( K ∩ aK ) it follows that A − 1 = A . F or every element x 0 ∈ X , the orbit of x 0 is the set h x 0 i = { g x 0 : g ∈ G } = Gx 0 . The or bits o f elements o f X partition the s et X . Let G \ X denote the set of o rbits of the group a ction, a nd define the function π : X → G \ X by π ( x ) = h x i . W e call G \ X the quotient sp ac e o f X by G , and we call π the quotient map of X onto G \ X . Note that ev ery orbit h x i is a subset of the set X and a point in the quotien t space G \ X . W e define the quo tient topolog y on G \ X as follows: A set V in G \ X is op en if and only if π − 1 ( V ) is o p en in X . This is the lar g est top olog y o n the quotient space G \ X such that the quotient map π is contin uous. W e call the gr oup action G y X c o-c omp act if the quo tient space G \ X is compact. An isometric, prop erly discontin uous , co - compact action o f a gr oup G on a b oundedly compact g eo desic metric space is ca lled a ge ometric actio n . W e now state the “ fundament al obser v ation of geometric g roup theory .” Theorem 5 . L et ( X , d ) b e a b ounde d ly c omp act ge o desic metric sp ac e and let G b e a gr oup that acts isometric al ly on X . Supp ose t hat the gr oup action G y X is pr op erly disc ontinuous and c o-c omp act. L et π : X → G \ X b e the quotient map, and let K b e a c omp act subset of X such that π ( K ) = G \ X . Then A = { a ∈ G : K ∩ a K 6 = ∅ } is a fi n ite set of gener ators for G . F or example, the additive gro up Z n of n -dimensional lattice points ac ts on E u- clidean space R n by transla tio n: α g ( x ) = g + x for g ∈ Z n and x ∈ Z n . The gr oup Z n acts isometrica lly on R n since | α g ( x ) − α g ( y ) | = | ( g + x ) − ( g + y ) | = | x − y | . 8 MEL VYN B. NA THANSON Let K b e a c o mpact subset of R n . Then K is bo unded and there is a num ber r > 0 such that | x | < r for all x ∈ K . If g ∈ Z n and K ∩ ( g + K ) 6 = ∅ , then there exists x ∈ K such that g + x ∈ K . Therefor e, | g | − r < | g | − | x | ≤ | g + x | < r and | g | < 2 r . Ther e exist only finitely man y lattice points in Z n of leng th less than 2 r , and so the actio n on Z n on R n is pro per ly discontin uous. W e shall prove that the gro up action Z n y R n is co -compact. Let π : R n → Z n \ R n be the quotient map. The quotien t space T n = Z n \ R n is called the n - dimensional torus . Let { W i } i ∈ I be an op en cov er of T n , a nd define V i = π − 1 ( W i ) for all i ∈ I . Then { V i } i ∈ I is an o p en cover o f R n . The unit cu b e K = [0 , 1] n = { x = ( x 1 , . . . , x n ) ∈ R n : 0 ≤ x i ≤ 1 for all i = 1 , . . . , n } is a compa ct subset of R n , a nd π ( K ) = T n . Since { V i } i ∈ I is an op en cov e r of K , it follows that there is a finite subset J of I such tha t K ⊆ S j ∈ J V j , and so T n = π ( K ) ⊆ [ j ∈ J π ( V j ) = [ j ∈ J W j . Therefore, T n is compa c t and the gro up action Z n y R n is co-compa c t. References [1] P . de la Harpe, T opics in Ge ometric Gr oup The ory , Chicago Lectures in Mathema tics, Uni - v ersity of Chicago Press, Chicago, IL, 2000. [2] V . A. Efremo viˇ c, The pr oximity ge ometry of Riemannian manifolds , Usp ekhi Mat. Nauk 8 (1953), 189. [3] J. M i lnor, A note on curvatur e and fundamental gr oup , J. Differen tial Geometry 2 (196 8), 1–7. [4] M . B. Nathanson, Phase tra nsitions in infinite ly gener ate d gr oups, and r elate d pr oblems in additive numb er the ory , arXiv: 0811.3990, 2008. [5] , Ne ts in gr oups, minimum length g -adic r epr esentations, and minimal additive c om- plements , arXiv: 0812.0560, 2008. [6] , Bi-Lipschitz e quivalent met rics on gr oups, and a pr oblem in additive numb er the ory , arXiv: 0902.3254, 2009. [7] A . S. ˇ Sv arc, A v olume invariant of c overings , Dokl. Ak ad. Nauk SSSR (N.S.) 105 (1955), 32–34. Dep ar tment of Ma thema tics, Lehman College (CUNY), Bronx, New York 10468 E-mail addr e ss : melvyn .nathanson @lehman.cuny.edu