Arithmetic progressions in self-similar sets

Abstract

Given a sequence {\{b_i\}}_{i=\mathrm{1}}^n and a ratio \lambda \in (\mathrm{0},\mathrm{1}), let E={\cup }_{i=\mathrm{1}}^n(\lambda E+b_i) be a homogeneous self-similar set. In this paper, we study the existence and maximal length of arithmetic progressions in E: Our main idea is from the multiple β-expansions.

Keywords

Self-similar sets / arithmetic progression (AP) / β-expansions

References

[1] Akiyama S, Komornik V. Discrete spectra and Pisot numbers. J Number Theory, 2013, 133(2): 375–390

[2] Broderick R, Fishman L, Simmons D. Quantitative results using variants of Schmidt's game: dimension bounds, arithmetic progressions, and more. Acta Arith, 2019, 188(3): 289–316

[3] Chaika J. Arithmetic progressions in middle-nth Cantor sets. arXiv: 1703.08998

[4] Dajani K, de Vries M. Invariant densities for random β-expansions. J Eur Math Soc (JEMS), 2019, 9(1): 157–176

[5] Dajani K, Jiang K, Kong D, Li W. Multiple codings for self-similar sets with overlaps. arXiv: 1603.09304

[6] Dajani K, Jiang K, Kong D, Li W. Multiple expansions of real numbers with digits set {0，1，q}. Math Z, 2019, 291(3-4): 1605–1619

[7] Dajani K, Kraaikamp C, van der Wekken N. Ergodicity of N-continued fraction expansions. J Number Theory, 2013, 133(9): 3183–3204

[8] De Vries M, Komornik V. Unique expansions of real numbers. Adv Math, 2009, 221(2): 390–427

[9] Erdös P, Turán P. On some sequences of integers. J Lond Math Soc, 1936, 11(4): 261–264

[10] Falconer K. Fractal Geometry: Mathematical Foundations and Applications. Chichester: John Wiley & Sons, Ltd, 1990

[11] Fraser J M, Yu H. Arithmetic patches, weak tangents, and dimension. Bull Lond Math Soc, 2018, 50(1): 85–95

[12] Furstenberg H, Katznelson Y, Ornstein D. The ergodic theoretical proof of Szemerédi's theorem. Bull Amer Math Soc (N S), 1982, 7(3): 527–552

[13] Glendinning P, Sidorov N. Unique representations of real numbers in non-integer bases. Math Res Lett, 2001, 8(4): 535–543

[14] Green B, Tao T. The primes contain arbitrarily long arithmetic progressions. Ann of Math (2), 2008, 167(2): 481–547

[15] Hutchinson J E. Fractals and self-similarity. Indiana Univ Math J, 1981, 30(5): 713–747

[16] Komornik V, Kong D, Li W. Hausdorff dimension of univoque sets and Devil's staircase. Adv Math, 2017, 305:165–196

[17] Laba I, Pramanik M. Arithmetic progressions in sets of fractional dimension. Geom Funct Anal, 2009, 19(2): 429–456

[18] Li J, Wu M, Xiong Y. On Assouad dimension and arithmetic progressions in sets defined by digit restrictions. J Fourier Anal Appl, 2019, 25(4): 1782–1794

[19] Roth K F. On certain sets of integers. J Lond Math Soc, 1953, 28: 104–109

[20] Shmerkin P. Salem sets with no arithmetic progressions. Int Math Res Not IMRN, 2017, (7): 1929–1941

[21] Sidorov N. Expansions in non-integer bases: lower, middle and top orders. J Number Theory, 2009, 129(4): 741–754

[22] Szemerédi E. On sets of integers containing no four elements in arithmetic progression. Acta Math Hungar, 1969, 20: 89–104

[23] Szemerédi E. On sets of integers containing no k elements in arithmetic progression. Acta Arith, 1975, 27: 199–245

[24] Tao T. What is good mathematics? Bull Amer Math Soc (N S), 2007, 44(4): 623–634

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