Fourier dimension and avoidance of linear patterns

Abstract

The results in this paper are of two types. On one hand, we construct sets of large Fourier dimension that avoid nontrivial solutions of certain classes of linear equations. In particular, given any finite collection of translation-invariant linear equations of the form(0.1)∑i=1vmixi=m0x0, with (m0,m1,⋯,mv)∈Nv+1,m0=∑i=1vmi and v≥2, we find a Salem set E⊆[0,1] of dimension 1 that contains no nontrivial solution of any of these equations; in other words, there does not exist a vector (x0,x1,⋯,xv)∈Ev+1 with distinct entries that satisfies any of the given equations. Variants of this construction can also be used to obtain Salem sets that avoid solutions of translation-invariant linear equations of other kinds, for instance, when the collection of linear equations to be avoided is uncountable or has irrational coefficients. While such constructions seem to suggest that Salem sets can avoid many configurations, our second type of results offers a counterpoint. We show that a set in R whose Fourier dimension exceeds 2/(v+1) cannot avoid nontrivial solutions of all equations of the form (0.1). In particular, a set of positive Fourier dimension must contain a nontrivial linear pattern of the form (0.1) for some v, and hence cannot be rationally independent. This is in stark contrast with known results [36] that ensure the existence of rationally independent sets of full Hausdorff dimension. The latter class of results may be viewed as quantitative evidence of the structural richness of Salem sets of positive dimension, even if the dimension is arbitrarily small.Not surprisingly, the proofs follow two distinct paths, depending on whether the result establishes existence or avoidance of patterns. The new methodological contribution of this article lies in the former, and involves constructing a family of pattern-identifying measures for E that cannot vanish simultaneously. The proofs of the avoidance results use classical constructions of discrete avoiding sets combined with a recent construction of Shmerkin [43], extending these tools to a more general framework.