Large subsets of Euclidean space avoiding infinite arithmetic progressions

Bradford, Laurestine; Kohut, Hannah; Mooroogen, Yuveshen

doi:10.1090/proc/16404

Large subsets of Euclidean space avoiding infinite arithmetic progressions
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by Laurestine Bradford, Hannah Kohut and Yuveshen Mooroogen;

Proc. Amer. Math. Soc. 151 (2023), 3535-3545

DOI: https://doi.org/10.1090/proc/16404

Published electronically: April 28, 2023

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Abstract:

It is known that if a subset of $\mathbb {R}$ has positive Lebesgue measure, then it contains arbitrarily long finite arithmetic progressions. We prove that this result does not extend to infinite arithmetic progressions in the following sense: for each $\lambda$ in $[0,1)$, we construct a subset of $\mathbb {R}$ that intersects every interval of unit length in a set of measure at least $\lambda$, but that does not contain any infinite arithmetic progression.

References

F. A. Behrend, On sets of integers which contain no three terms in arithmetical progression, Proc. Nat. Acad. Sci. U.S.A. 32 (1946), 331–332. MR 18694, DOI 10.1073/pnas.32.12.331
Alex Burgin, Samuel Goldberg, Tamás Keleti, Connor MacMahon, and Xianzhi Wang, Large sets avoiding infinite arithmetic/geometric progressions, arXiv:2210.09284 (2022).
Miroslav Chlebík, On the Erdős similarity problem, arXiv:1512.05607 (2015).
Jacob Denson, Malabika Pramanik, and Joshua Zahl, Large sets avoiding rough patterns, Harmonic analysis and applications, Springer Optim. Appl., vol. 168, Springer, Cham, [2021] ©2021, pp. 59–75. MR 4238597, DOI 10.1007/978-3-030-61887-2_{4}
Robert Fraser and Malabika Pramanik, Large sets avoiding patterns, Anal. PDE 11 (2018), no. 5, 1083–1111. MR 3785600, DOI 10.2140/apde.2018.11.1083
Tamás Keleti, Construction of one-dimensional subsets of the reals not containing similar copies of given patterns, Anal. PDE 1 (2008), no. 1, 29–33. MR 2431353, DOI 10.2140/apde.2008.1.29
Mihail N. Kolountzakis and Effie Papageorgiou, Large sets containing no copies of a given infinite sequence, arXiv:2208.02637 (2022).
Péter Maga, Full dimensional sets without given patterns, Real Anal. Exchange 36 (2010/11), no. 1, 79–90. MR 3016405, DOI 10.14321/realanalexch.36.1.0079
András Máthé, Sets of large dimension not containing polynomial configurations, Adv. Math. 316 (2017), 691–709. MR 3672917, DOI 10.1016/j.aim.2017.01.002
Joe Repka, Personal communication, 2022.
Pablo Shmerkin, Salem sets with no arithmetic progressions, Int. Math. Res. Not. IMRN 7 (2017), 1929–1941. MR 3658188, DOI 10.1093/imrn/rnw097
R. E. Svetic, The Erdős similarity problem: a survey, Real Anal. Exchange 26 (2000/01), no. 2, 525–539. MR 1844133, DOI 10.2307/44154057
E. Szemerédi, On sets of integers containing no $k$ elements in arithmetic progression, Acta Arith. 27 (1975), 199–245. MR 369312, DOI 10.4064/aa-27-1-199-245
Hermann Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313–352 (German). MR 1511862, DOI 10.1007/BF01475864
Alexia Yavicoli, Large sets avoiding linear patterns, Proc. Amer. Math. Soc. 149 (2021), no. 10, 4057–4066. MR 4305962, DOI 10.1090/proc/13959
Samuel S. Wagstaff Jr., Sequences not containing an infinite arithmetic progression, Proc. Amer. Math. Soc. 36 (1972), 395–397. MR 308074, DOI 10.1090/S0002-9939-1972-0308074-6