New bounds for dimensions of a set uniformly avoiding multi-dimensional arithmetic progressions
aa r X i v : . [ m a t h . C A ] O c t NEW BOUNDS FOR DIMENSIONS OF A SETUNIFORMLY AVOIDING MULTI-DIMENSIONALARITHMETIC PROGRESSIONS
KOTA SAITO
Abstract.
Let r k ( N ) be the largest cardinality of a subset of { , . . . , N } whichdoes not contain any arithmetic progressions (APs) of length k . In this paper, wegive new upper and lower bounds for fractal dimensions of a set which does notcontain ( k, ε )-APs in terms of r k ( N ), where N depends on ε . Here we say that asubset of real numbers does not contain ( k, ε )-APs if we can not find any APs oflength k with gap difference ∆ in the ε ∆-neighborhood of the set. More precisely,we show multi-dimensional cases of this result. As a corollary, we find equivalencesbetween multi-dimensional Szemer´edi’s theorem and bounds for fractal dimensionsof a set which does not contain multi-dimensional ( k, ε )-APs. introduction A real sequence ( a j ) k − j =0 is called an arithmetic progression of length k if thereexists ∆ > a j = a + ∆ j for all j = 0 , , . . . , k −
1. We say ∆ is the gap difference of ( a j ) k − j =0 . It is abig problem to show the existence or non-existence of arithmetic progressions in agiven set. Recently, we get great progresses on the problem. For example, Greenand Tao proved that the set of prime numbers contains arbitrarily long arithmeticprogressions [10].Let us define an arithmetic patch which is a higher dimensionalized arithmeticprogression. Let v = { v , . . . , v m } be a set of orthogonal unit vectors in R d where1 ≤ m ≤ d . For every k ∈ N and ∆ >
0, we say that a set P ⊂ R d is an arithmeticpatch (AP) of size k and scale ∆ with respect to orientation v if P = ( t + ∆ m X i =1 x i v i : x i = 0 , , . . . , k − ) for some t ∈ R d . For every ε ∈ [0 , / Q ⊂ R d is a ( k, ε, v ) -AP if thereexists an arithmetic patch P of size k , and scale ∆ > Mathematics Subject Classification.
Primary: 11B25, 28A80.
Key words and phrases. fractal dimensions, Assouad dimension, arithmetic progressions, arith-metic patches, multi-dimensional Szemer´edi’s theorem.The author announced the one dimensional cases of the results of this paper in the conferencestitled ‘Research on the Theory of Random Dynamical Systems and Fractal Geometry’ in KyotoUniversity on 31st August, 2019, and ‘RIMS Workshop 2019, Analytic Number Theory and RelatedTopics’ in Kyoto University on 18th October, 2019. v such that(1.1) sup x ∈ P inf y ∈ Q k x − y k ≤ ε ∆ . Note that ( k, , v )-APs are arithmetic patches of size k with orientation v . Fraserand Yu gave the original notion of ( k, ε, v )-APs in [7]. The term ( k, ε, v )-APs wasfirstly seen in [5]. The existence of ( k, ε, v )-APs of a given set F is connected withthe Assouad dimension of F . Fraser and Yu showed that a subset of R d has Assouaddimension d if and only if the set contains ( k, ε, v )-APs for every k ≥ ε >
0, andbasis v . Here the orthogonality of v does not require in their paper and they considernot only R d but also any finitely dimensional Banach spaces. Note that Fraser andYu say that F aymptotically contains arbitrarirly large arithemetic patches in [7]instead that F contains ( k, ε, e )-APs for every k ≥ ε > e denotes somefixed basis on a finitely dimensional Banach space. Furthermore, Fraser, the author,and Yu gave the quantitative upper bound of the Assouad dimension of a subset of R d which does not contain ( k, ε, v )-APs as follows: Theorem 1.1 ([5, Theorem 5.1]) . Fix integers m and d with ≤ m ≤ d , and fix k ≥ and ε ∈ (0 , / √ d ) . Let F ⊆ R d . If F does not contain ( k, ε, v ) -APs for somea set of orthogonal unit vecters v = { v , . . . , v m } , then we have dim A F ≤ d + log(1 − /k m )log( k ⌈√ d/ (2 ε ) ⌉ ) . We now define D A ( k, ε, d, m ) = sup { dim A F : F ⊆ R d , F does not contain any ( k, ε, v )-APs(1.2) for some a set of orthogonal unit vectors { v , . . . , v m }} , and D A ( k, ε ) = D A ( k, ε, , . We also define D H ( k, ε, d, m ) by replacing dim A and F ⊆ R d in (1.2), to dim H and the condition that F ⊂ R d is compact. Here dim H F denotes the Hausdorffdimension of F . By Theorem 1.1, we obtain the upper bound for D A ( k, ε, d, m ). Inparticular, when d = m = 1, Fraser, the author and Yu give lower and upper boundsfor D A ( k, ε ) and D H ( k, ε ) in [5] as follows:(1.3) log 2log k − − εk − − ε ≤ D H ( k, ε ) ≤ D A ( k, ε ) ≤ − /k )log k ⌈ / (2 ε ) ⌉ for every k ≥ ε ∈ (0 , /
2) with ε < ( k − / k, ε, v )-APs for some set of orthogonal unit vectors { v , . . . , v m } , in terms of the function r k,m ( N ). Here r k,m ( N ) denote the largestcardinality of A ⊆ { , . . . , N } m such that A does not contain any arithmetic patchesof size k with orientation { e , . . . , e m } , where e i denotes the vector in R d of which i -thcoordinate is 1 and others are 0. Further, we give the equivalent conditions betweenmulti-dimensional Szemer´edi’s theorem given by Furstenberg and Katznelson [4] andbounds for D A ( k, ε, d, d ). EAK APS AND MULTI-DIMENSIONAL SZEMER’EDI’S THEOREM 3
Notation 1.2.
We give the following notations: • N denotes the set of all positive integers; • for every F ⊆ R d , dim L F denotes the lower dimension of F , dim P F denotesthe packing dimension of F , dim LB F denotes the lower box dimension of F ,and dim UB F denotes the upper box dimension of F ; • for every X ∈ { L , H , P , LB , UB } , define D X ( k, ε, d, m ) = sup { dim X F : F ⊂ R d is compact , F does not contain any( k, ε, v )-APs for some set of orthogonal unit vectors { v , . . . , v m }} , and D X ( k, ε ) = D X ( k, ε, , . • for every x ∈ R , ⌈ x ⌉ denotes the minimum integer n such that x ≤ n , and ⌊ x ⌋ denotes the maximum integer n such that x ≥ n ; • for every finite set A , | A | denotes the cardinality of A .2. Result
Theorem 2.1.
Fix integers k ≥ , d ≥ and ≤ m ≤ d , and fix a real number ε ∈ (0 , / . If F ⊆ R does not contain any ( k, ε, v ) -APs for some set of orthogonalunit vectors v = { v , . . . , v m } , then we have dim A F ≤ inf N ∈ N log( ⌈√ d/ε ⌉ d N d − m r k,m ( N ))log( N ⌈√ d/ε ⌉ ) . In particular, if we substitute N = ⌈√ d/ε ⌉ , then (2.1) dim A F ≤ d + 12 log( r k,m ( ⌈√ d/ε ⌉ ) / ⌈√ d/ε ⌉ m )log ⌈√ d/ε ⌉ . We will prove Theorem 2.1 in Section 4. This gives a better upper bound for D A ( k, ε, d, m ) by the multidimensional Szemer´edi’s theorem if ε is sufficiently small.This will be claimed in Corollary 2.5. Theorem 2.2.
Fix integers k ≥ , d ≥ and ≤ m ≤ d , where k ≥ when d = 1 .Fix a real number < ε < / . Let N = ⌈ / (8 ε ) ⌉ , < δ ≤ / and A be a subsetof { , , . . . , N − } d which does not contain any arithmetic patches of size k withorientation { e , . . . , e m } . For all a ∈ A and x ∈ R , we define φ a ( x ) = δN − δ x + a. Let F be the attractor of the iterated function system ( φ a ) a ∈ A , that is, F = [ a ∈ A φ a ( F ) . Then the following hold: (i) the iterated function system { φ a : a ∈ A } satisfies open set condition; (ii) F does not contain any ( k, ε, { e , . . . , e m } ) -APs; (iii) it follows that dim H F = log | A | log (cid:0) N − δ + 1 (cid:1) . K. SAITO
We will prove Theorem 2.2 in Section 4. This theorem gives a new lower boundfor D A ( k, ε, d, m ). Here a set of contractive functions { f , . . . , f n } from R d to R d iscalled an iterated function system on R d . We say that an iterated function system { f , . . . , f n } on R d satisfies open set condition if there exists a bounded open set V ⊂ R d such that V ⊇ n [ i =1 f i ( V ) , where the union on the left hand side is pairwise disjoint. The open set condition isuseful to calculate the Hausdorff dimension (see [2, H] ). We now define D S ( k, ε, d, m ) = { dim H F : F ⊂ R d is compact and satisfies (i) and (ii) in Theorem 2.2 } . for every k ≥
2, 0 < ε < / ≤ m ≤ d . Note that for every bounded set F ⊆ R d , we have(2.2) dim L F ≤ dim H F ≤ dim P F ≤ dim LB F ≤ dim UB F ≤ dim A F. Further, by Fraser’s result [3], if F satisfies (i) in Theorem 2.2, then we have(2.3) dim L F = dim H F = dim P F = dim LB F = dim UB F = dim A F. Therefore we can replace dim H in the definition of D S ( k, ε, d, m ) by dim X for all X ∈ { L , P , LB , UB , A } . We refer [2, 3, 12] to the readers who are interested in moredetails on fractal dimensions. Corollary 2.3.
For every d ≥ and k ≥ where k ≥ when d = 1 , and for every < ε < / , one has d (cid:18) − log 32log(4 /ε ) (cid:19) + log( r k,m ( ⌈ / (8 ε ) ⌉ ) / ⌈ / (8 ε ) ⌉ m )log(4 /ε ) ≤ D S ( k, ε, d, m ) ≤ D A ( k, ε, d, m ) ≤ d + 12 log( r k,m ( ⌈√ d/ε ⌉ ) / ⌈√ d/ε ⌉ m )log ⌈√ d/ε ⌉ . We will prove Corollary 2.3 in Section 3 by combining Theorem 2.1 and Theo-rem 2.2. Recently, in [6], Fraser, Shmerkin and Yavicoli define d ( k, ε ) = sup { dim H F : F ⊂ R is a bounded set which does not contain ( k, ε, { } )-APs } . They prove that d ( k, ε ) = sup { dim H F : F ⊂ R does not contain ( k, ε, { } )-APs } = sup { dim A F : F ⊂ R is a bounded set which does not contain ( k, ε, { } )-APs }} . Therefore D H ( k, ε ) = d ( k, ε ) ≤ D A ( k, ε ). Further, they give upper and lower boundsfor d ( k, ε ) as follows:(2.4) log r k, ( ⌊ / (10 ε ) ⌋ )log(10 ⌊ / (10 ε ) ⌋ ) ≤ d ( k, ε ) ≤ (cid:18) log( r k, ( ⌈ /ε ⌉ ) + 1)log ⌈ /ε ⌉ + 12 (cid:19) . These bounds are almost same as the bounds in Corollary 2.3 with d = m = 1. EAK APS AND MULTI-DIMENSIONAL SZEMER’EDI’S THEOREM 5
Corollary 2.4.
Fix any < δ < . For every k ≥ , D A ( k, ε ) is less than or equalto (2.5) 1 − c k (L ( ε ) − L ( ε ) − δ )(1 + exp( − L ( ε ) − δ ))L ( ε ) + exp( − exp(L ( ε )(1 − L ( ε ) − δ ))) for all < ε < ε ( δ ) , where we define L ( ε ) = log ⌈ /ε ⌉ , L n ( ε ) = log L n − ( ε ) forevery n ≥ . We will prove Corollary 2.5 in Section 3. The first term in the numerator of thecomplicated fraction in (2.5) dominates the second, and also the first term of thedenominator dominates the second. Hence the right hand side of (2.5) is near to1 − c k L ( ε )(1 + exp( − L ( ε ) − δ ))L ( ε )Therefore we obtain better upper bounds if δ > r k, ( N ) [8] as follows:(2.6) r k, ( N ) ≤ N (log log N ) c k for every N ≥ k ≥
3, where c k = 2 − k +9 . In order to simplify, substitute δ = 1 / D A ( k, ε ) ≤ − (1 + o (1)) c k log log log ⌈ /ε ⌉ log ⌈ /ε ⌉ as ε → +0. This upper bound is better than (1.3) if 0 < ε < ε ( k ) is sufficientlysmall. Corollary 2.5.
For every k ≥ and < ε < / , D S ( k, ε ) is greater than or equalto − ε ) (cid:18) log(32 C ) + (log 2) (cid:18) n ( n − / n p log ⌈ / (8 ε ) ⌉ + 12 n log log ⌈ / (8 ε ) ⌉ (cid:19)(cid:19) , for some absolute constant C > , where n = ⌈ log k ⌉ . This result immediately comes from Corollary 2.3 with d = m = 1 and O’Bryant’slower bound for r k, ( N ) [11], which is r k, ( N ) ≥ CN exp (cid:18) (log 2) (cid:18) − n ( n − / n p log N + 12 n log log N (cid:19)(cid:19) for all N ≥ k ≥
3, for some
C >
0. Hence we omit the proof. In order tosimplify, for any fixed k ≥ < ε < ε ( k ) we have D S ( k, ε ) ≥ − A k n p log ⌈ / ε ⌉ log(1 /ε )for some constant A k > k . This lower bound is better than(1.3) if 0 < ε < ε ( k ) is sufficiently small. K. SAITO
Corollary 2.6.
For every d ≥ , there exist positive constants A d and B d such thatfor every integer k ≥ and real number < ε < / , one has A d r k,d ( ⌈ / ε ⌉ ) ⌈ / ε ⌉ d ≤ ε d − D S ( k,ε,d,d ) ≤ ε d − D H ( k,ε,d,d ) ≤ ε d − D A ( k,ε,d,d ) ≤ B d r k,d ( ⌈√ d/ε ⌉ ) ⌈√ d/ε ⌉ d ! / . We can immediately show this corollary from Corollary 2.3 with d = m . Thuswe omit the proof. This corollary gives the following equivalences between themultidimensional Szemer´edi’s theorem given by Furstenberg and Katznelson, andbounds for D ( k, ε, d, d ): Corollary 2.7.
Fix integers k ≥ and d ≥ . Let e be the standard basis of R d .The following are equivalent: (i) If A ⊆ N d satisfies lim N →∞ | A ∩ [1 , N ] d | N d > , then A contains ( k, , e ) -APs; (ii) r k,d ( N ) /N → as N → ∞ ; (iii) ε d − D S ( k,ε,d,d ) → as ε → +0 ; (iv) ε d − D H ( k,ε,d,d ) → as ε → +0 ; (v) ε d − D A ( k,ε,d,d ) → as ε → +0 . We will prove Corollary 2.7 in Section 3. Furstenberg and Katznelson proved thatfor any A ⊆ Z d satisfying(2.7) lim h →∞ sup (cid:26) | A ∩ I | h d : I ⊂ R d is a closed hyper-cube with side length h (cid:27) > , and for any finite set F ⊂ Z d , there exists a ∈ Z d and ∆ ∈ Z such that a + ∆ F ⊂ A .This statement is equivalent to (i) in Corollary 2.7 is true for every d ≥ k ≥
3. Further, D H ( k, ε, d, d ) in (iv) in Corollary 2.7 can be replaced other fractaldimensions. Therefore for each X ∈ { S , L , H , P , LB , UB , A } , the multidimensionalSzemer´edi’s theorem is equivalent tolim ε → +0 ε d − D X ( k,ε,d,d ) = 0for every k ≥ d ≥
1. 3.
Proof of Corollaries
Let d and m be integers with 1 ≤ m ≤ d , and let k ≥ N ≥
1, we define r k,d,m ( N ) as the largest cardinality of A ⊆ { , · · · , N } d such that A does not contain any arithmetic patches of size k with orientation { e , . . . , e m } . Lemma 3.1.
For every ≤ m ≤ d , k ≥ , and N ≥ , we have r k,d,m ( N ) = N d − m r k,m ( N ) . EAK APS AND MULTI-DIMENSIONAL SZEMER’EDI’S THEOREM 7
Proof.
Let A ⊆ { , . . . , N } d which does not contain ( k, , { e , . . . , e m } )-APs and | A | = r k,m ( N ). Define B = { ( x, y ) ∈ { , . . . , N } m × { , . . . , N } d − m : x ∈ A, y ∈ { , . . . , N } d − m } Then B is a subset of { , . . . , N } d and does not contain ( k, , v )-APs. Therefore onehas r k,d,m ( N ) ≤ | B | = N d − m r k,m ( N ) . On the other hand, take any A ⊆ { , · · · , N } d which does not contain any ( k, ε, e )-APs. For every 1 ≤ j m +1 , . . . , j d ≤ N , define B j m +1 , ··· ,j d = { x ∈ { , . . . , N } m : ( x, j m +1 , . . . , j d ) ∈ A } . Then each B j m +1 ,...,j d does not contain any ( k, ε, e )-APs. Hence we obtain N d − m r k,m ( N ) ≤ X j m +1 ,...,j d | B j m +1 ,...,j d | = | A | = r k,d,m ( N ) . (cid:3) Proof of Corollary 2.3.
By (2.1) in Theorem 2.1, one has D A ( k, ε, d, m ) ≤ d + 12 log( r k,m ( ⌈√ d/ε ⌉ ) / ⌈√ d/ε ⌉ m )log ⌈√ d/ε ⌉ . for all k ≥
2, 0 < ε < / ≤ m ≤ d . We next find a lower bound for D S ( k, ε, d, m ). Let N = ⌈ / (8 ε ) ⌉ . Take B ⊆ { , . . . , N } d which does not contain( k, , { e , . . . , e m } )-APs and | B | = r k,d,m ( N ). By Lemma 3.1 and Theorem 2.2 with A = B and δ = 1 /
24, one has D S ( k, ε, d, m ) ≥ log r k,d,m ( ⌈ / (8 ε ) ⌉ )log(24( ⌈ / (8 ε ) ⌉ −
1) + 1) ≥ log( r k,m ( ⌈ / (8 ε ) ⌉ ) ⌈ / (8 ε ) ⌉ d − m )log(4 /ε ) ≥ d (cid:18) − log 32log(4 /ε ) (cid:19) + log( r k,m ( ⌈ / (8 ε ) ⌉ ) / ⌈ / (8 ε ) ⌉ m )log(4 /ε ) . (cid:3) Proof of Corollary 2.5.
Fix 0 < δ < k ≥ < ε ≪ δ N = ⌈⌈ /ε ⌉ r ⌉ where r ( ε ) = exp( − (log log ⌈ /ε ⌉ ) − δ ) and δ = 1 /
2. By Theorem 2.1 with d = m = 1, we have D A ( k, ε, ≤ log( r k ( N ) ⌈ /ε ⌉ )log( N ⌈ /ε ⌉ ) ≤ − c k log log( r log ⌈ /ε ⌉ )( r + 1) log ⌈ /ε ⌉ + log(1 + ⌈ /ε ⌉ − r ) ≤ − c k log(L ( ε ) − L ( ε ) − δ )(1 + exp( − L ( ε ) − δ ))L ( ε ) + ⌈ /ε ⌉ − r , which implies that D A ( k, ε ) is less than or equal to(3.1) 1 − c k (L ( ε ) − L ( ε ) − δ )(1 + exp( − L ( ε ) − δ ))L ( ε ) + exp( − exp(L ( ε )(1 − L ( ε ) − δ ))) . (cid:3) Proof of Corollary 2.7.
By Corollary 2.6, (ii)-(v) are equivalent. Thus it suffices toshow that (i) and (ii) are equivalent. The following lemma implies this equivalence:
K. SAITO
Lemma 3.2.
Fix k ≥ and d ≥ , and let e be the standard basis on R d . Thefollowing are equivalent: (i) r k,d ( N ) /N d → as N → ; (ii) Any A ⊆ Z d with (2.7) contains ( k, , e ) -APs; (iii) If A ⊆ N d satisfies (3.2) lim N →∞ | A ∩ [1 , N ] d | N d > , then A contains ( k, , e ) -APs. We prove this lemma in Appendix. (cid:3) Proof of main Theorems
For every x ∈ R d and R > B ( x, R ) denotes the closed ball with radius R centered at x ∈ R d . For every bounded set E ⊂ R d and r > N ( E, r ) denotes thesmallest cardinality of a family of sets whose diameters are less than or equal to r .The Assouad dimension of F ⊆ R d is defined bydim A F = inf n σ ≥ ∃ C > ∀ r > ∀ R > r ∀ x ∈ FN (cid:16) B ( α, R ) ∩ F, r (cid:17) ≤ CR σ o . By this definition, we obtain that for every F ⊆ R d dim A F = inf n σ ≥ ∃ C > ∃ λ ≥ ∀ r > ∀ R > λr ∀ x ∈ F (4.1) N (cid:16) B ( α, R ) ∩ F, r (cid:17) ≤ CR σ o . Proof of Theorem 2.1.
Choose any set F which does not contain ( k, ε, { v , . . . , v m } )-APs. By rotating, we may assume that v = e , . . . , v m = e m . Suppose that √ d/ε is an integer. Fix any small real number α and large parameter λ = λ ( α ), and fixany r, R with R/r > λ . Fix a ball B of R d with radius R and centered at a point in F . Choose a hyper-cube C ⊇ B with side length 2 R . Write C = d Y i =1 [ a i , a i + 2 R ] . Fix any positive integer N . For every i = 1 , , . . . , d and j = 0 , , . . . , √ dN/ε − A ( i ) j = [ a i + 2 jRε/ ( N √ d ) , a i + 2( j + 1) Rε/ ( N √ d )] . Let c ( i ) j be the middle point of A ( i ) j for all i and j . Let us find a family of hyper-cubeswith side length 2 Rε/ ( N √ d ) which covers F ∩ B and whose cardinality is less thanor equal to(4.2) √ dε ! d r k,d,m ( N ) . EAK APS AND MULTI-DIMENSIONAL SZEMER’EDI’S THEOREM 9
Here define I ( j , . . . , j d ) = { ( j + √ dn /ε, . . . , j d + √ dn d /ε ) ∈ Z d : 0 ≤ n , . . . , n d ≤ N − } for every 1 ≤ j , . . . , j d ≤ √ d/ε . Note that [ ≤ j ,...,j d ≤√ dN/ε I ( j , . . . , j d ) = { , , . . . , √ dN/ε } d , which is a disjoint union. Fix any 1 ≤ j , . . . , j d ≤ √ d/ε . Let I = I ( j , . . . , j d ) , P = { ( c (1) x , . . . , c ( d ) x d ) ∈ R d : ( x , . . . , x d ) ∈ I } , S = ( d Y i =1 A ( i ) x i : ( x , . . . , x d ) ∈ I ) . Assume that the number of A ∈ S such that F ∩ A = ∅ is at least r k,d,m ( N ) + 1.Then we can find an arithmetic patch Q ⊆ P of size k and scale ∆ ≥ R/N withorientation { e , . . . , e m } satisfying that for all x ∈ Q , there exists y = y ( x ) ∈ F suchthat k x − y k ≤ Rε/N ≤ ε ∆ . Thus { y ( x ) : x ∈ Q } is a ( k, ε, e )-AP. This is a contradiction. Hence the numberof A ∈ S such that F ∩ A = ∅ is less than or equal to r k,d,m ( N ) for each fixed1 ≤ j , . . . , j d ≤ √ d/ε . Therefore one has (4.2). We iterate this argument t -timesfor each smaller hyper-cubes which intersect F . Here t is a positive integer which isdetermined later. Then the number of hyper-cubes with side length 2 R ( ε/ ( N √ d )) t which covers F is less than or equal to ( √ d/ε ) dt r k,d,m ( N ) t . Let t = & log(2 R √ d/r )log( N √ d/ε ) ' . Then one has 2 R √ d (cid:18) εN √ d (cid:19) t ≤ R √ d (cid:18) εN √ d (cid:19) log( r/ (2 R √ d ))log( ε/ ( N √ d )) = r. Therefore we obtain that N ( F ∩ B, r ) ≤ N ( F ∩ C, r ) ≤ √ dε ! d r k,d,m ( N ) t ≤ R √ dr ! (1+ α ) log(( √ d/ε ) drk,d,m ( N ))log( N √ d/ε ) . Hence by (4.1), we conclude thatdim A F ≤ (1 + α ) log(( √ d/ε ) d r k,d,m ( N ))log( N √ d/ε ) , which implies that dim A F ≤ log(( √ d/ε ) d r k,d,m ( N ))log( N √ d/ε )as α → +0.If √ d/ε is not an integer, then let ε ′ = √ d ⌈√ d/ε ⌉ . It is seen that ε ′ ≤ ε . Therefore F does not contain ( k, ε ′ , { v , . . . , v m } )-APs forsome a set of orthogonal unit vectors { v , . . . , v m } . Hence one hasdim A F ≤ log( ⌈√ d/ε ⌉ d N d − m r k,m ( N ))log( N ⌈√ d/ε ⌉ )by Lemma 3.1 and the assumption that √ d/ε ′ is an integer. (cid:3) Proof of Theorem 2.2.
Let A ⊆ { , . . . , N − } d be a set which does not contain anyarithmetic patches of size k with orientation { e , . . . , e m } . Define φ a ( x ) = δN − δ x + a ( a ∈ A, x ∈ R d ) ,I = [0 , N − δ ] d , I n +1 = [ a ∈ A φ a ( I n ) ( n ≥ , F = ∞ \ n =1 I n . Then it follows that I n ⊇ I n +1 for every n ≥
0. In fact, for all ( x , . . . , x d ) ∈ I and( a , . . . , a d ) ∈ A , one has0 ≤ δN − δ x i + a i ≤ N − δ, which means that I ⊆ I . If I n +1 ⊆ I n holds for some n ≥
0, then we have I n +2 ⊆ [ a ∈ A φ a ( I n ) = I n +1 . The set F is the attractor of { φ a : a ∈ A } since if F ′ denotes the attractor of { φ a : a ∈ A } , then by the triangle inequality and the monotonicity of ( I n ) n ≥ , onehas d H ( F, F ′ ) ≤ d H ( I n , F ) + d H ( I n , F ′ ) ≤ d H ( I n , F ) + (cid:18) δN − (cid:19) n d H ( I , F ′ ) → +0as n → ∞ . Here d H ( A, B ) denotes the Hausdorff metric between compact sets A and B of R d . Therefore F ′ = F . The iterated function system { φ a : a ∈ A } satisfiesopen set condition since one has(0 , N − δ ) d ⊇ [ a ∈ A φ a ((0 , N − δ ) d ) , and the union on the right hand side is disjoint. This yields thatdim H F = log | A | log( N − δ + 1) EAK APS AND MULTI-DIMENSIONAL SZEMER’EDI’S THEOREM 11 by Hutchinson’s theorem (alternatively see [2, Theorem 9.3]). The remaining part isto show that F does not contain ( k, ε, { e , . . . , e m } )-APs. Let e = { e , . . . , e m } . As-sume that F contains a ( k, ε, { e , . . . , e m } )-APs. Let Q be such a ( k, ε, { e , . . . , e m } )-APs. It suffices to show that(4.3) Q ⊆ φ a ( I )for some a ∈ A . If (4.3) is true, then φ − a ( Q ) ⊆ I and φ − a ( Q ) is also a ( k, ε, e )-AP.Thus there exists a ∈ A such that φ − a ( Q ) ⊆ φ a ( I )which implies that φ − a ◦ φ − a ( Q ) ⊆ I . We iterate this argument t -times for anypositive integer t . Then there exists a , . . . , a t ∈ A such that Q ⊆ φ a ◦ · · · ◦ φ a t ( I ) . The diameter of the right hand side goes to 0 as t → ∞ . This is a contradiction.Let us show that (4.3). By the definition of F , Q ⊆ I . Hence for all x ∈ Q thereexists a ( x ) ∈ A such that(4.4) k a ( x ) − x k ∞ ≤ δ. Here for every x = ( x , . . . , x d ) ∈ R d , k x k denotes the Euclidean norm and k x k ∞ =max {| x i | : 1 ≤ i ≤ d } . By definition, there exists ∆ > k andscale ∆ with orientation e such that(4.5) inf y ∈ P k x − y k ∞ ≤ inf y ∈ P k x − y k ≤ ε ∆for all x ∈ Q . Let y ( x ) be the point y ∈ P which satisfies (4.5) for every x ∈ Q .Here recall that Q ⊆ I = S a ∈ A φ a ( I ). Fix any x ∈ Q and choose x ′ ∈ Q such that k y ( x ) − y ( x ′ ) k ∞ = ∆where x and x ′ are distinct. Then by (4.4) and (4.5), one has |k a ( x ′ ) − a ( x ) k ∞ − ∆ | ≤ k a ( x ′ ) − y ( x ′ ) − ( a ( x ) − y ( x )) k ∞ ≤ k a ( x ′ ) − x ′ k ∞ + k x ′ − y ( x ′ ) k ∞ + k a ( x ) − x k ∞ + k x − y ( x ) k ∞ ≤ δ + ε ∆) . Since Q ⊆ I , one has N − δ ≥ ( k − − ε ∆ , which implies that ∆ ≤ N − δk − ε . Since k ≥
2, 0 < ε < / < δ ≤ / |k a ( x ′ ) − a ( x ) k ∞ − ∆ | ≤ (cid:18) δ + ⌈ / (8 ε ) ⌉ − δk − − ε ε (cid:19) (4.6) ≤ (cid:18) δ + 1 / (8 ε ) + δ − / ε (cid:19) < / a ( x ′ ) , a ( x ) ∈ Z d , k a ( x ′ ) − a ( x ) k ∞ ∈ Z . Therefore by (4.6), k a ( x ) − a ( x ′ ) k is aconstant which does not depend on x or x ′ , which implies that { a ( x ) : x ∈ Q } is an AP of size k with orientation e . This is a contradiction. Hence at least two points x, x ′ ∈ Q belong to φ a ( I ) for some a ∈ A . This yields that∆ ≤ δ + 2 ε ∆ , which implies that(4.7) ∆ ≤ δ − ε . Take x ′′ ∈ Q \ { x, x ′ } such thatdist( { x, x ′ } , Q \ { x, x ′ } ) = dist( { x, x ′ } , { x ′′ } ) , where dist( A, B ) = inf {k x − y k : x ∈ A, y ∈ B } for every A, B ⊆ R d . Thus by (4.7),one has min {k x − x ′′ k , k x ′ − x ′′ k} ≤ (1 + 2 ε )∆ ≤ δ < − δ. Therefore x ′′ does not reach to other islands φ a ′ ( I ) ( a ′ ∈ A \ { a } ), which meansthat x ′′ must belong to φ a ( I ). By replacing { x, x ′ } to { x, x ′ , x ′′ } , we can iteratethe same argument until the number of x ∈ Q such that x ∈ φ a ( I ) reaches | Q | .Therefore we get Q ⊆ φ a ( I ). (cid:3) Discrete Analogue
For every F ⊆ N , defineDim ζ F = lim N →∞ log | F ∩ [1 , N ] | log N = inf ( σ ≥ X n ∈ F n − σ < ∞ ) , which is introduced by Doty, Gu, Lutz, Mayordomo, and Moser in [1], and general-ized to a metric space by the author in [13]. We can see that(5.1) Dim ζ F ≤ dim A F for all F ⊆ N by the definition of the Assouad dimension. The author showed theinequality (5.1) more generally in [13]. Define D ζ ( k, ε ) = sup { Dim ζ F : F ⊆ N does not contain any ( k, ε, { } )-APs } . By Theorem 2.1 with d = m = 1, one has D ζ ( k, ε ) ≤ D A ( k, ε ) ≤
12 log( r k ( ⌈ /ε ⌉ ) ⌈ /ε ⌉ )log( ⌈ /ε ⌉ ) . Theorem 5.1.
Fix k ≥ and ε ∈ (0 , / . Let N = ⌈ / (8 ε ) ⌉ , η be an integer with η ≥ and A be a subset of { , , . . . , N − } with ∈ A which does not contain anyarithmetic progressions of length k . Define ψ a ( x ) = ( η + 1)( N − x + a ( a ∈ A, x ∈ Z ) B = { } , B n = [ a ∈ A ψ a ( B n − ) ( n ≥ , F = ∞ [ n =0 B n . Then the following hold:
EAK APS AND MULTI-DIMENSIONAL SZEMER’EDI’S THEOREM 13 (i) it follows that F ⊆ N ∪ { } and F = [ a ∈ A ψ a ( F );(ii) F does not contain any ( k, ε, { } ) -APs; (iii) it follows that lim N → + ∞ log | F ∩ [1 , N ] | log N ≥ log | A | log((1 + η )( N − . We can find a set A ⊆ { , , . . . , N − } with 0 ∈ A and | A | = r k, ( N ) since if A ⊆ { , , . . . , N − } does not contain any arithmetic progressions of length k , then( − min A ) + A is a subset of { , , . . . , N − } with 0 ∈ A which does not containarithmetic progression of length k . Therefore we have(5.2) log r k ( ⌈ / (8 ε ) ⌉ )log(1 /ε ) ≤ D ζ ( k, ε ) ≤ D A ( k, ε ) ≤ inf N ∈ N log( r k ( N ) ⌈ /ε ⌉ )log( N ⌈ /ε ⌉ )for every k ≥ < ε < /
16. Hence we get the following discrete analogue ofCorollary 2.7.
Corollary 5.2.
Fix k ≥ . Any A ⊆ N with positive upper density contains arith-metic progressions of length k if and only if lim ε → +0 ε − D ζ ( k,ε ) = 0 . Proof of Theorem 5.1.
We can easily show (i) in Theorem 5.1 by definition and thefact that B ⊆ B . Let us show that (iii). Let N ′ = N − ξ = ( η + 1) N ′ . Forall n ≥
1, it follows thatdiam( B n ) ≤ ξ diam( B n − ) + N ′ ≤ ξ diam B n − + ξN ′ + N ′ ≤ · · · ≤ ( ξ n − + ξ n − + · · · + 1) N ′ ≤ (6 / ξ n − N ′ . Hence one has | F ∩ [0 , (6 / ξ n − N ′ ] | ≥ | B n ∩ [0 , (6 / ξ n − N ′ ] | ≥ | A | n , since the union B n = [ a ∈ A ψ a ( B n − )is disjoint for every n ≥
1. Therefore we obtainlim N →∞ log | F ∩ [0 , N ]log N ≥ log | A | log((1 + η )( N − N → ∞ . Let us next show (ii). It follows that(5.3) B n = [ a ∈ A ( B n − + ξ n − a ) for all n ≥
1. This is clear when n = 1. Assume that (5.3) holds for some n ≥ B n +1 = [ a ∈ A ψ a ( B n ) = [ a ∈ A [ a ′ ∈ A ξB n − + ξ n a ′ ! + a ! = [ a ′ ∈ A [ a ∈ A η N B n − + a ! + ξ n a ′ ! = [ a ∈ A ( B n + ξ n a ) . Assume that F contains a ( k, ε, { } )-AP. Let P be such a ( k, ε, { } )-AP. Thenwe can find n ≥ P ⊆ B n . By a similar discussion of the proof ofTheorem 2.2, P ⊆ B n − + ξ n a for some a ∈ A . Hence B n − contains a ( k, ε, { } )-AP. By iterating this discussion, we conclude that B contains a ( k, ε, { } )-AP. Thisis a contradiction. (cid:3) Further discussion
Question 6.1.
Is it ture that D S ( k, ε ) ≤ − log(log ⌈ / (8 ε ) ⌉ (log log ⌈ / (8 ε ) ⌉ ) )log(1 /ε ) for every k ≥ and < ε < ε ( k ) ? Erd˝os-Tur´an conjecture states that a subset of positive integers whose sum ofreciplocals diverges would contain arbitrarily long arithmetic progressions. Thisconjecture is still open even if the length of arithmetic progressions is equal to 3. Bypartial summation, if for every k ≥
3, there exists C k > N ≥ r k ( N ) ≤ C k N log N (log log N ) , then Erd˝os-Tur´an conjecture would be ture (see [9]). Therefore by combining thisimplication and Corollary 2.6 with d = 1, the affirmative answer to Question 6.1implies the Erd˝os-Tur´an conjecture. Question 6.2.
Can we prove that lim ε → +0 ε − D X ( k,ε ) = 0 for all k ≥ for some X ∈ { ζ , L , H , P , LB , UB , A } , by using fractal geometry? By Corollary 2.7, the affirmative answer to Question 6.2 gives another proof ofSzemer´edi’s theorem [14].
Appendix A. Proof of Lemma 3.2
Proof.
Let us show that (i) implies (ii). Let δ be the left hand side of (2.7). Thenthere exist infinitely many hyper-cubes I , I , . . . such that h < h < · · · → ∞ , and | A ∩ I n | h dn > δ EAK APS AND MULTI-DIMENSIONAL SZEMER’EDI’S THEOREM 15 for all n ∈ N , where h n denotes the side length of I n . For sufficiently large n , wehave(A.1) | A ∩ I n | ≥ δ ⌈ h n ⌉ d > r k,d ( ⌈ h n ⌉ ) ⌈ h n ⌉ d ⌈ h n ⌉ d = r ( ⌈ h n ⌉ ) . Here there exists t ∈ Z d such that(A.2) t + A ∩ I n ⊆ { , . . . , ⌈ h n ⌉} d . By combing (A.1) and (A.2), t + A ∩ I n contains ( k, , e )-APs, which implies that A contains ( k, , e )-APs.It is clear that (ii) implies (iii). Therefore let us prove that (iii) implies (i). Thisis clear when d = 1 and k = 2. Thus we discuss the cases when d = 1 and k ≥ d ≥ k ≥
2. Fix d ≥
1. Assume that r k,d ( N ) /N d does not go to 0 as N → ∞ . Let us construct a subset of integers which satisfies (3.2) and does notcontain any ( k, , e )-APs. We find a positive real number δ and an infinite sequence N < N < · · · of integers such that r k,d ( N j ) > δN j for every j ∈ N . Then for every j ∈ N , choose A j ⊆ { , , . . . , N j } d which does notcontain any ( k, , e )-APs and | A j | = r k,d ( N ). Let t be the origin of R d , B = t + A and M = N . It is clear that B ⊂ [1 , M ]. Assume that we have an increasingsequence of sets B ⊂ B ⊂ · · · ⊂ B n and integers M < M < · · · < M n such that B n ⊂ [1 , M n ] d . Then take N j n +1 with N j n +1 > M n , and let t n +1 = ( M n + 2 N j n +1 ) e , B n +1 = B n ∪ ( t n +1 + A j n +1 ) , M n +1 = M n + 3 N j n +1 . We can find that B n ⊂ B n +1 , M n < M n +1 and B n +1 ⊂ [1 , M n +1 ] d . We iteratethis discussion inductively and get a sequence of sets B ⊂ B ⊂ · · · . Define B = ∪ n ∈ N B n . It follows that | B ∩ [1 , M n ] d | = | B n ∩ [1 , M n ] d | ≥ | A j n | > δN dj n ≥ δ d M dn , which means that B satisfies (3.2). Let us show that B does not contain any ( k, , e )-APs. Assume that B contains some ( k, , e )-AP, then let P be such a ( k, , e )-AP.There exists an integer r such that P ⊆ B r . By the choice of A N r , P ⊆ t r + A N r doesnot hold. If P intersects B r − and t r + A N r , then there exist two elements p, p ′ ∈ P such that p ∈ t r + A r , p ′ ∈ B r − , p ′ = p + ∆ e for some ∆ >
0. Then we have∆ ≥ t r = M r − + 2 N j r . Thus other terms of P do not belong to B r . This is a contradiction. Hence P ⊆ B r − .By iterating this discussion, we conclude that P ⊆ A , which is a contradiction.Therefore B does not contain any ( k, , e )-APs. (cid:3) Acknowledgement
The author would like to thank Professor Kohji Mastumoto and Professor MasatoMimura for useful comments. This work is supported by Grantin-Aid for JSPSResearch Fellow (Grant Number: 19J20878).
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