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RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:Szemerédi's theorem and fractal dimensions of setsavoiding (κ, ε)-arithmetic progressions (Research on theTheory of Random Dynamical Systems and FractalGeometry)Saito, KotaSaito, Kota. Szemerédi's theorem and fractal dimensions of sets avoiding (κ, ε)-arithmetic progressions (Research onthe Theory of Random Dynamical Systems and Fractal Geometry). 数理解析研究所講究録 2021, 2176: 102-1102021-04http://hdl.handle.net/2433/264789102Szemeredi's theorem and fractal dimensions of sets avoiding (k, E)-arithmetic progressions Kota Saito 1 Arithmetic progressions of integers We say that a real sequence (ai)に~ C 股 isan arithmetic progression (AP) of length k ifthere exists△ >0 such that 佑＝ ％ ＋i△ for i = 0, 1,..., k -1. We say that△is the gap length of an AP. We say that a set P is an AP of length k if there exists an AP (ai)にlof length k in the sense of a sequence such that P = { ai: i = 0, 1,..., k -1 }.It is a big problem to show the existence or non-existence of arithmetic progressions of a given set. One of main topics of this problem is to find long APs of a given 'large'set. For example, a subset A of positive integers with positive upper density i.e. IAn [1,N]I limsup N N→OO >0 contains arbitrarily long APs. This statement is known as Szemeredi's the-orem which was proved by Szemeredi in 1975 [14]. This theorem implies van der Waerden's theorem [15] which states that if we partition a set of al positive integers into finitely many sets, then we can find at least one set which contains arbitrarily long arithmetic progressions. Erdos and Turan conjectured the case when the length of APs is 3 of Szemeredi's theorem in [2]. In the same paper, they introduced the number rk(N) as the maximum cardinality of A such that Aこ{1,2,..., N} does not contain any APs of length k. We can get the equivalent statement of Szemeredi's theorem in terms of rk(N) as follows: Szemeredi's theorem is equivalent to lim rk(N)/N = 0 N→OO holds for al k 2 3. Furthermore, if N rk(N) ＜ --'log N(log N)2 103was true for al k 2: 3 and N 2: N(k), then Erdos-Turan conjecture would be true. We can see a proof of this implication in [7]. Here Erdos-Thran conjecture states that a subset A of positive integers satisfying 1 -= 00 こa aEA would contain arbitrarily long arithmetic progressions. This conjecture is stil open even if the length of APs is equal to 3. Note that the sum ofreciprocals of prime numbers is divergent. Therefore, if Erdos-Turan conjecture is true, we can prove the Green-Tao theorem [8] which states that the set of al prime numbers contains arbitrarily long arithmetic progressions. Hence it is important to find non-trivial bounds for rk(N). The result by Gowers in the paper [6] is the best in known upper bounds for rk(N) for general length k, that is, for al N 2: 3 and k 2:3, rk(N) ＜ N ―(log log N)2-2 k+9・(1) Remark that we know better upper bounds for rk(N) in the particular case when k = 3 or 4 (see [1, 9]). O'Bryant gave a lower bound for rk(N). For al N ~ 2 and k ~ 3, we have 1 叫N)2: CN exp ((log 2) (-n2(n-l)/2砂~+犀og2 log2 N)), (2) for some C > 0, where n =「log2kl By combining (1) and (2), for every k 2:3, there exists positive constants Ck and Dk such that for every N > e尺we have N exp(-Ck(log Ntl2) ::;叫N)::;Nexp(-D且oglog log N). 2 Arithmetic progressions of real numbers We firstly construct a Cantor set which does not contain any APs of length 3. Let心1(x)= x/6 andゅ2(x)= x/6 + 5/6. Let Ii。=［0, 1], and define In=灼（In-1)U応(In-1)for al n 2: 1. Then we define C = n:=。In.Note that C is the attractor of the iterated function system｛叫心｝ on股． Weshow that C does not contain any APs of length 3 by contradiction. If C contained at least one 104AP of length 3, then let P be such an AP, and Pこf1 by definition. The set 1 is the union of two disjoint closed intervals. The middle hole of 1 is enough large such that al of elements in P belong to the left or right interval. Therefore there exists i1 E { 1,2} such that Pこi/Ji1(Io) (3) By P ~ 12 and (3), there exists i2 E {1, 2} such that PC四 0訟 (Io).By iterating these steps, the diameter of P can be arbitrarily small. This is a contradiction. The iterated function system｛叫，砂｝ clearlysatisfies open set condition. Therefore the Hausdorff dimension of C is equal to real number s such that 6―s+6―s = 1. log2 Thus we have dimH C = ~ by Hutchinson's theorem [10], where dimH F log6 denotes the Hausdorff dimension of F. In particular, C is uncountable since the Hausdorff dimension of C is positive. Therefore, we can not characterized the existence of APs by using the cardinality of a given set. Question 1. Is it true that a set of real numbers with Hausdorff dimension one contains arbitrarily long arithmetic progressions? We may guess that the answer to Question 1 is"yes" because of Ramsey-type phenomena like Szemeredi's theorem and van der Waerden's theorem. Surprisingly, the answer is no. It is due to Keleti in 1998 [1]. He proved that there exists a compact set A of real numbers with Hausdorff dimension one such that the solutions x, y, z, w EA to the equation X + y = z + w, X < Y, z < w (4) are only x = z and y = w. If y = z in the equation (4), sets {x, w, y} are APs of length 3. Therefore A does not contain APs of length 3. Therefore, the existence of APs can not be characterized by the Hausdorff dimension. 3 Weak arithmetic progressions For al 0こE< 1/2, we say that a real sequence (ai)に~ is a (k, E)-AP if there exists an arithmetic progression (b』に~ of length k with gap length△ >。such that lai -bilこe△for al i = 0, 1,..., k -1. For example, when k = 5, E = 1/4, the following sequence of points near to bi's is an (5, 1/4)-AP: 105b △ b b2 b b | •い·い·(戸下ミ．．し三•4• | }△ The original notion of (k, E)-APs can be seen in [5]. In the literature, the term "(k, E)-APs" was firstly appeared in [4]. The existence of (k, E)-APs can be caracterized by the Assouad dimension. Here for al bounded sets Ee配 andfor al positive real number r, we define N(E, r) as the minimum cardinality of a family of sets with diameter less than or equal to r which covers E. For al Fこ記 wedefine dimげ＝ inf{ <7 ~ 0 : there exists C > 0 such that for al O < r < R and x E F, we have N(B(x, R) n F, r) :; C(り）び｝which is called the Assouad dimension of F. Here B(x, R) denotes the closed ball of配 withradius R centered at x. By the result of Fraser and Yu [5], a subset F of real numbers contains (k, E)-APs for al k ~ 3 and EE (0, 1/2) if and only if dimA F = l. Further, they proved that a subset of positive integers whose sum ofreciprocals is divergent contains (k, E)-APs for al k ~ 3 and EE (0, 1/2) [5]. Fraser, the author, and Yu gave quantitative bounds for the Assouad (Hausdorff) dimension of a set which does not contain (k, E)-APs for fixed k and E. More precisely, for every F c股， k~ 3, and E E (0, 1/2), if F does not contain any (k, E)-APs, then we have dimAF ~ 1 + log(l -1/k) log(k 「1/ （2€)l) ．(5) Note that for every set F ~良叫 the following inequality holds: dimLFさdimHF ~ <limp F ~ dimA F, where dimL F denotes that the lower dimension of F, and dimp F denotes that the packing dimension of F. We can see this formula in [3]. Hence one can replace dimA in (5) by dimx for al X E {L, H, P}. Remark that if F C 配 isbounded, then the upper box dimension of F is less than or equal to the Assouad dimension of it. Thus we can also replace the Assouad dimension by the upper box dimension in (5) if we restrict that Fis bounded. Now define DA(k, s) := sup{ dimA F I Fこ罠 doesnot contain (k, E)-APs}. 106We can consider広 (k,E) as an analogue of rk(N). The right hand side of (5) is an upper bound for広 (k,E). In order to find effective lower bounds, we should construct a subset of real numbers which has large dimension and does not contain (k, E)-APs. Fraser, the author, and Yu also gave a lower bound for DA(k, E). For al k 2: 3 and E E (0, 1) satisfying Eく (k-2)/4, there exists a set F c罠 whichdoes not contain any (k, E)-APs and dimげ＝ dimHF = log2 log 2k-2-4c:. k-2-4c: (6) Now define加 (k,E) := sup{ dimH F I Fこ尺 doesnot contain (k,E)-APs}. By combining (5), (6), and the inequality dimH F :=:; dimげ， wehave log2 log(1 -1/k) さ応(k,E) さ DA(k,E):s;l+~- (7) log 2k-2-4c log (k「1/（2e)l)．k-2-41o 4 Main results This section gives the results which the author states in the conference These results are one dimensional cases in the paper [13]. Theorem 2. For every k 2'.3 and O < c: < 1/2, we have 広 (K,e)＜!log(rk(「1/el)「1/el)-2 log(「1/c:l).By Gowers'upper bounds, we have 広（K,€）さ 1-1 log log log「1/cl2l+2K+9 log「1/cl (8) This upper bound is better than (5) if c issufficiently small. Actually, if 0 < c < exp(-expexpexp(2 + 2k+9)), then one has (8) < (5). Theorem 3. Fix k ~ 3 and O < c < 1/16. Then we can construct a compact set F c尺whichsatisfies the following conditions: (i) there exists a set of linear contractions {f1,..., fm} with open set con-dition such that F is the attractor of {Ji,..., f m} i.e. m F = LJJi(F); i=l (i) F does not contain any (k, E)-APs; 107(ii) dim□ 2 logrk(ll/(8E)」）log(l/E) ・ Here f:股→股 iscalled a linear contraction if f (x) = ax + b for some lal < 1. Define Ds(k, c) = sup{ dimH F I F satisfies (i) and (i) in Theorem 3}. Then we have :; Ds(k, c) ::;恥(k,E)::;DA(k,E)三一log rk(l 1/ (8E)」） llog(rk(「1/cl)「1/cl) log(l/E) 2 log(「1/cl) ・ By O'Bryant's lower bound for rk(N), we get a better lower bound for Ds(k, E) for sufficiently small O < E < E(k). Corollary 4. There exists positive real numbers C and D such that For every O < E < 1/16 and k 2'.3, we have 叫1/ （8€)」）< €1-Ds(K,€)< €1-DH(K,€) < €1-DA(K,€) < E rk(「1／叶）1/2 C ll/(8E)」- - -（「1/€l)Hence for fixed k ~ 3. It is known that a subset of positive integers with positive upper density contains arithmetic progressions of length k if and only if limN→00 rk(N)/N = 0. Therefore from Corollary 4, the following are equivalent: (i) a subset of positive integers with positive upper density contains arith-metic progressions of length k; (i) limN→oo rk(N)/N = O; (ii) limE→+0 El-Dx(k,E) = 0 for some X E {S, H, A}; (iv) limE→+0 El-Dx(k,E) = 0 for al XE {S, H, A}. Actually, by Szemeredi's theorem, (i),(i),(ii), and (iv) are true. If one showed (ii) for al k ~ 3, then we would get another proof of Szemeredi's theorem. Remark that by (5), implies that 1-DA(k,E) log(l-1/k)) c < c logk「1/（2c)l= （ log(l/c) 1 -1/k)logk「1/（2c)l →1-1/k as E→ +0. Thus the upper bound given by Fraser, the author, and Yu does not reach to prove Szemeredi's theorem. 1085 Sketch of proofs Sketch of Proof of Theorem 2. Fix any k 2 3 and E E (0, 1/2). Take any F~ 恥 which does not contain (k, E)-APs. Take any interval I= [a, a+ 2R] centered at a point in F. Fix any positive integer N. We partition I into N / E small closed intervals Aj with diameter 2RE/ N as follows: Aj =[a+ 2jRc/N, a+ 2(j + l)Rc/N] for al j = 0, l,..., N / E -1. For any 0さi< 1/E, let T(i)={i+j/c:Oさj::SN-1}.Fix O :; i < 1 / E.Since F does not contain any arithmetic progressions of length k, the number of j E T(i) such that Aj n Fヂ0is less than or equal to rk(N). Hence the number of j E {O, 1,..., N/E -1} such that F n Ajヂ0is less than or equal to r(N)/c. We iterate this argument t-times for each smaller itervals which intersect F, where t isa real number such that E ¥ t 2R(~r =r. Therefore we obtain that N(FnI,r）こ（rk(€N) ） t こ（誓）亨齊)Therefore we conclude that dimげ<log(rk(N)/c) log(N/c) ・ ロSketch of Proof of Theorem 3. Fix any k 2 3. Let N = ll/(8E)」． LetA be a subset of {O, 1,..., N -1} with IAI = rk(N) which does not contain arithmetic progressions of length k. Let c5 = 1/16 and define X 叫(x)= ~+a (a EA), N-l+c5 (9) I。=［0, N -1 + c5], In = LJ叫 In-1) (n 2 1), (10) aEA and define F = n:=o In. Then F is the attractor of the iterated function sysytem { 1Pa: a E A} which satisfies open set condition. Thus one has dimげこlog rk(l 1/ (Sc)」）log(l/c) ・ By contradiction, we can see that F does not contain (k, E)-APs. ロWe refer [13] to the readers who want to know more details. 109Acknowledgment This work was supported by JSPS KAKENHI Grant Numbers 19J20878. References [1] T. F. Bloom. A quantitative improvement for Roth's theorem on arith-metic progressions, J. London Math. Soc. 93 (2016), 643-663. [2] P. Erdos and P. Turan. On some sequences of integers, J. London Math. Soc. 11 (1936), 261-264. [3] J. M. Fraser. Assouad type dimensions and homogeneity of fractals, Trans. Amer. Math. Soc., 366 (2014), 6687-6733. [4] J. M. Fraser, K. Saito and H. Yu. Dimensions of sets which uni-formly avoid arithmetic progressions. Int. Math. Res. Not. IMRN., 2017, rnx261. [5] J. M. Fraser and H. Yu. Arithmetic patches, weak tangents, and dimen-sion, Bull. Lond. Math. Soc.,50 (2018), 85-95. [6] W.T. Gowers. A new proof of Szemeredi's theorem, Geom. Funct. Anal., 11 (2001), 465-588. [7] W.T. Gowers. Erdos and arithmetic progressions, Erdos Centennial, Bolyai Society Mathematical Studies, 25, L. Lovasz, I. Z. Ruzsa, V. T. Sos eds., Springer 2013, 265-287. [8] B. Green and T. Tao. The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2), 167 (2008), 481-547. [9] B. Green and T. Tao. New bounds for Szemeredi's theorem, II: a poly-logarithmic bound for r4(N), Mathematika 63 (2017), 944-1040. [10] J. E. Hutchinson. Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. [1] T. Keleti, A 1-dimensional subset of the reals that intersects each of its translates in at most a single point, Real Anal. Exchange, 24:2 (1998), 843-844. [12] K. O'Bryant. Sets of integers that do not contain long arithmetic pro-gressions, Electron. J. Combin., 18 (2011), Paper 59, 15 pp. 110[13] K. Saito. New bounds for dimensions of a set uniformly avoiding multi-dimensional arithmetic progressions, preprint, 2019, available at https://arxiv.org/abs/1910.13071. [14] E. Szemeredi. On sets of integers containing no k elements in arithmetic progression, Collection of articles in memory of Jurii Vladimirovic Lin-nik. Acta Arith., 27 (1975), 199-245. [15] B. L. van der Waerden. Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk. 15 (1927), 212-216. 

Szemerédi's theorem and fractal dimensions of sets avoiding (κ, ε)-arithmetic progressions (Research on the Theory of Random Dynamical Systems and Fractal Geometry)

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Szemerédi's theorem and fractal dimensions of sets avoiding (κ, ε)-arithmetic progressions (Research on the Theory of Random Dynamical Systems and Fractal Geometry)

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