On Sets Containing an Affine Copy of Bounded Decreasing Sequences
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(2020) 26:73
On Sets Containing an Affine Copy of Bounded Decreasing Sequences Tongou Yang1 Received: 20 January 2019 / Revised: 5 December 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract How small can a set be while containing many configurations? Following up on earlier work of Erd˝os and Kakutani (Colloq Math 4:195–196, 1957), Máthé (Fund Math 213(3):213–219, 2011) and Molter and Yavicoli (Math Proc Camb Soc 168:57–73, 2018), we address the question in two directions. On one hand, if a subset of the real numbers contains an affine copy of all bounded decreasing sequences, then we show that such a subset must be somewhere dense. On the other hand, given a collection of convergent sequences with prescribed decay, there is a closed and nowhere dense subset of the reals that contains an affine copy of every sequence in that collection. Keywords Sparse sets containing pattern · Dimension · Density Mathematics Subject Classification 11B05 · 28A78 · 28A12 · 28A80
1 Introduction Given sets A, B ⊆ R, we say that A contains the pattern B if A contains an affine copy of B, i.e. if there exist δ = 0 and t ∈ R such that t + δ B ⊆ A. Identification of patterns in sets is an active research area, and there are questions of many flavours: (1) Which types of patterns are guaranteed to exist in large sets? For example, a classical consequence of the Lebesgue density theorem is that if E ⊆ R has positive Lebesgue measure, then it contains an affine copy of all finite sets. In sets of fractal dimensions, Łaba and Pramanik [16] proved that if a fractal set A supports a measure satisfying a Frostman’s condition and has sufficiently large
Communicated by Alex Iosevich.
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Tongou Yang [email protected] Department of Mathematics, The University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T1Z2, Canada 0123456789().: V,-vol
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Journal of Fourier Analysis and Applications
(2020) 26:73
Fourier decay, then A must contain a 3-term arithmetic progression. Other results in this direction include [1,9,12,17]. (2) Can there exist large sets avoiding prescribed patterns? A famous conjecture in this direction is the Erd˝os similarity problem (see [5]), which is stated as follows: for each infinite set S ⊆ R, does there exist a measurable set E with positive Lebesgue measure that does not contain any affine copy of S? There are partial results to this conjecture by Bourgain, Falconer, Kolountzakis, etc; see [2,7,15]. Apart from the Erd˝os similarity conjecture, there are also lots of well-known results about large sets avoiding patterns. Keleti [13] showed that for any set A ⊆ R of at least 3 elements there exists a set of Hausdorff dimension 1 that contains no similar copy of A. In this direction, Shmerkin [20] showed that there exists a set of Fourier dimension 1 that contains no 3-term arithmetic progression. In another direction, Fraser and Pramanik [8] obtained a general result that there exist sets of large Hausdorff dimension and full Minkowski dimension that avoid
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