Dimensions of Sets Which Uniformly Avoid Arithmetic Progressions

Abstract

Abstract. We provide estimates for the dimensions of sets in R which uniformly avoid finite arithmetic progressions. More precisely, we say F uniformly avoids arithmetic progressions of length k ≥ 3 if there is an ε > 0 such that one cannot find an arithmetic progression of length k and gap length ∆ > 0 inside the ε∆ neighbourhood of F . Our main result is an explicit upper bound for the Assouad (and thus Hausdorff) dimension of such sets in terms of k and ε. In the other direction, we provide examples of sets which uniformly avoid arithmetic progressions of a given length but still have relatively large Hausdorff dimension.We also consider higher dimensional analogues of these problems, where arithmetic progressions are replaced with arithmetic patches lying in a hyperplane. As a consequence we obtain a discretised version of a 'reverse Kakeya problem': we show that if the dimension of a set in R d is sufficiently large, then it closely approximates arithmetic progressions in every direction. Almost arithmetic progressions and dimensionArithmetic progressions are fundamental objects across mathematics and conditions which either force them to exist (or not exist) within a given set are of particular interest. For example, Szemerédi's seminal theorem [Sz] states that if A ⊂ N has positive upper density, then A contains arbitrarily long arithmetic progressions. We say a set {a i } k−1 i=0 ⊂ R is an arithmetic progression (AP) of length k if there exists ∆ > 0 such that a i = a 0 + i∆, for i = 1, 2, . . . , k − 1. We say ∆ is the gap length of the arithmetic progression. We are primarily interested in sets which uniformly avoid arithmetic progressions and for this reason it is useful to introduce a weaker notion of 'almost arithmetic progressions'. In particular, given ε ≥ 0 we say that {b i } k−1 i=0 ⊂ R is a (k, ε)-AP if there exists an arithmetic progression {a i } k−1 i=0 with gap length ∆ > 0 such thatfor all i = 0, 2, . . . , k − 1. Thus there is an arithmetic progression of length k and gap length ∆ > 0 inside the closed ε∆ neighbourhood of a (k, ε)-AP. We think of a set F ⊂ R as uniformly avoiding arithmetic progressions of length k if, for some ε > 0, it does not contain any (k, ε)-APs. Note that (k, 0)-APs are simply the usual arithmetic progressions of length k.

Assouad Dimension and Fractal Geometry

“…, x + (k − 1)∆}, for some x ∈ R, that is, a collection of k points each separated from the next by a common distance ∆. We consider the following weakening of the property of containing arithmetic progressions, which was introduced and studied in [110,107]. This definition should be understood as saying that, for arbitrarily large k and arbitrarily small ε > 0, X gets within ε of an arithmetic progression of length k. The fact that ε∆ appears instead of ε is the necessary normalization.…”

“…One may bound the Assouad dimension of a set above in terms of how far away it is from approximating arithmetic progressions of a given length. This is explored in detail in [107]. A quantitative improvement of Theorem 14.1.3 was proved in [113] where the error o(∆) is replaced with O(∆ α ) for any α ∈ (0, 1).…”

New dimension spectra: Finer information on scaling and homogeneity

“…Also, Hieronymi and Miller have recently used the Assouad dimension to study nondefinability problems relating to expansions in the real number field, see [HM]. There are also connections between the Assouad dimension and problems in arithmetic combinatorics, for example the existence of arithmetic progressions or asymptotic arithmetic progressions, see [FY2,FSY].…”

Dimension growth for iterated sumsets

“…If a set is Ahlfors-David regular, then the lower, Hausdorff, box and Assouad dimensions all coincide and, as such, our results below apply to a much larger class of sets where Ahlfors-David regularity is weakened to only requiring that either the lower dimension is strictly positive or the Assouad dimension is strictly less than 1. This is natural since, for example, sets with Assouad dimension strictly less than 1 are precisely the sets which uniformly avoid arithmetic progressions [FY18,FSY17], and arithmetic progressions tend to cause the sumset to be small.…”