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 \begin{equation} \sum_{i=1}^v m_ix_i=m_0x_0, \; \text{ with } (m_0, m_1, \cdots, m_v) \in \mathbb N^{v+1}, m_0 = \sum_{i=1}^{v} m_i \text{ and } v \geq 2, \label{rational-eqn} \end{equation} we find a Salem set $E \subseteq [0,1]$ of dimension 1 that contains no nontrivial solution of any of these equations; in other words, there does not exist a vector $(x_0, x_1, \cdots, x_v) \in E^{v+1}$ 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 $\mathbb R$ whose Fourier dimension exceeds $2/(v+1)$ cannot avoid nontrivial solutions of all equations of the above form. In particular, a set of positive Fourier dimension must contain a nontrivial linear pattern of the above form for some $v$, and hence cannot be rationally independent. This is in stark contrast with known results \cite{M17} 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.