Arithmetic progressions in self-similar sets

doi:10.1007/s11464-019-0788-2

Front. Math. China

2019, Vol. 14

Issue (5) : 957-966 https://doi.org/10.1007/s11464-019-0788-2

RESEARCH ARTICLE

Arithmetic progressions in self-similar sets

Lifeng XI(

), Kan JIANG, Qiyang PEI

Department of Mathematics, Ningbo University, Ningbo 315211, China

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Abstract

Given a sequence ${b i} i = 1 n$ and a ratio $λ ∈ (0, 1)$ , let $E = ∪ i = 1 n (λ E + b i)$ 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

Corresponding Author(s): Lifeng XI

Issue Date: 22 November 2019

Cite this article:

Lifeng XI,Kan JIANG,Qiyang PEI. Arithmetic progressions in self-similar sets[J]. Front. Math. China, 2019, 14(5): 957-966.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-019-0788-2
https://academic.hep.com.cn/fmc/EN/Y2019/V14/I5/957

1	S Akiyama, V Komornik. Discrete spectra and Pisot numbers. J Number Theory, 2013, 133(2): 375–390 https://doi.org/10.1016/j.jnt.2012.07.015
2	R Broderick, L Fishman, D Simmons. Quantitative results using variants of Schmidt's game: dimension bounds, arithmetic progressions, and more. Acta Arith, 2019, 188(3): 289–316 https://doi.org/10.4064/aa171127-8-11
3	J Chaika. Arithmetic progressions in middle-nth Cantor sets. arXiv: 1703.08998
4	K Dajani, M de Vries. Invariant densities for random β-expansions. J Eur Math Soc (JEMS), 2019, 9(1): 157–176 https://doi.org/10.4171/JEMS/76
5	K Dajani, K Jiang, D Kong, W Li. Multiple codings for self-similar sets with overlaps. arXiv: 1603.09304
6	K Dajani, K Jiang, D Kong, W Li. Multiple expansions of real numbers with digits set {0，1，q}. Math Z, 2019, 291(3-4): 1605–1619 https://doi.org/10.1007/s00209-018-2123-0
7	K Dajani, C Kraaikamp, N van der Wekken. Ergodicity of N-continued fraction expansions. J Number Theory, 2013, 133(9): 3183–3204 https://doi.org/10.1016/j.jnt.2013.02.017
8	M De Vries, V Komornik. Unique expansions of real numbers. Adv Math, 2009, 221(2): 390–427 https://doi.org/10.1016/j.aim.2008.12.008
9	P Erdös, P Turán. On some sequences of integers. J Lond Math Soc, 1936, 11(4): 261–264 https://doi.org/10.1112/jlms/s1-11.4.261
10	K Falconer. Fractal Geometry: Mathematical Foundations and Applications. Chichester: John Wiley & Sons, Ltd, 1990 https://doi.org/10.2307/2532125
11	J M Fraser, H Yu. Arithmetic patches, weak tangents, and dimension. Bull Lond Math Soc, 2018, 50(1): 85–95 https://doi.org/10.1112/blms.12112
12	H Furstenberg, Y Katznelson, D Ornstein. The ergodic theoretical proof of Szemerédi's theorem. Bull Amer Math Soc (N S), 1982, 7(3): 527–552 https://doi.org/10.1090/S0273-0979-1982-15052-2
13	P Glendinning, N Sidorov. Unique representations of real numbers in non-integer bases. Math Res Lett, 2001, 8(4): 535–543 https://doi.org/10.4310/MRL.2001.v8.n4.a12
14	B Green, T Tao. The primes contain arbitrarily long arithmetic progressions. Ann of Math (2), 2008, 167(2): 481–547 https://doi.org/10.4007/annals.2008.167.481
15	J E Hutchinson. Fractals and self-similarity. Indiana Univ Math J, 1981, 30(5): 713–747 https://doi.org/10.1512/iumj.1981.30.30055
16	V Komornik, D Kong, W Li. Hausdorff dimension of univoque sets and Devil's staircase. Adv Math, 2017, 305:165–196 https://doi.org/10.1016/j.aim.2016.03.047
17	I Laba, M Pramanik. Arithmetic progressions in sets of fractional dimension. Geom Funct Anal, 2009, 19(2): 429–456 https://doi.org/10.1007/s00039-009-0003-9
18	J Li, M Wu, Y Xiong. On Assouad dimension and arithmetic progressions in sets defined by digit restrictions. J Fourier Anal Appl, 2019, 25(4): 1782–1794 https://doi.org/10.1007/s00041-018-9641-3
19	K F Roth. On certain sets of integers. J Lond Math Soc, 1953, 28: 104–109 https://doi.org/10.1112/jlms/s1-28.1.104
20	P Shmerkin. Salem sets with no arithmetic progressions. Int Math Res Not IMRN, 2017, (7): 1929–1941 https://doi.org/10.1093/imrn/rnw097
21	N Sidorov. Expansions in non-integer bases: lower, middle and top orders. J Number Theory, 2009, 129(4): 741–754 https://doi.org/10.1016/j.jnt.2008.11.003
22	E Szemerédi. On sets of integers containing no four elements in arithmetic progression. Acta Math Hungar, 1969, 20: 89–104 https://doi.org/10.1007/BF01894569
23	E Szemerédi. On sets of integers containing no k elements in arithmetic progression. Acta Arith, 1975, 27: 199–245 https://doi.org/10.4064/aa-27-1-199-245
24	T Tao. What is good mathematics? Bull Amer Math Soc (N S), 2007, 44(4): 623–634 https://doi.org/10.1090/S0273-0979-07-01168-8

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