Large Salem Sets Avoiding Nonlinear Configurations

Abstract

We construct large Salem sets avoiding patterns, complementing previous constructions of pattern avoiding sets with large Hausdorff dimension. For a (possibly uncountable) family of uniformly Lipschitz functions $\{f_i:({\mathbb{T}}^d{)}^{n-2}\to {\mathbb{T}}^d\}$, we obtain a Salem subset of ${\mathbb{T}}^d$ with dimension $d/(n-1)$ avoiding nontrivial solutions to the equation $x_n-x_{n-1}=f_i(x_1,\dots ,x_{n-2})$. For a countable family of smooth functions $\{f_i:({\mathbb{T}}^d{)}^{n-1}\to {\mathbb{T}}^d\}$ satisfying a modest geometric condition, we obtain a Salem subset of ${\mathbb{T}}^d$ with dimension $d/(n-3/4)$ avoiding nontrivial solutions to the equation $x_n=f(x_1,\dots ,x_{n-1})$. For a set $Z\subset {\mathbb{T}}^{dn}$ which is the countable union of a family of sets, each with lower Minkowski dimension $s$, we obtain a Salem subset of ${\mathbb{T}}^d$ of dimension $(dn-s)/(n-1/2)$ whose Cartesian product does not intersect $Z$ except at points with non-distinct coordinates.

39 pages, 1 figure