Sets in ${\mathbb{R}}^d$ with slow-decaying density that avoid an unbounded collection of distances

Abstract

For any $d\in \mathbb{N}$ and any function $f:(0,\infty )\to [0,1]$ with $f(R)\to 0$ as $R\to \infty$, we construct a set $A\subseteq {\mathbb{R}}^d$ and a sequence $R_n\to \infty$ such that $\Vert x-y\Vert \ne R_n$ for all $x,y\in A$ and $\mu (A\cap B_{R_n})\ge f(R_n)\mu (B_{R_n})$ for all $n\in \mathbb{N}$, where $B_R$ is the ball of radius $R$ centered at the origin and $\mu$ is Lebesgue measure. This construction exhibits a form of sharpness for a result established independently by Furstenberg-Katznelson-Weiss, Bourgain, and Falconer-Marstrand, and it generalizes to any metric induced by a norm on ${\mathbb{R}}^d$.

3 pages, minor typos corrected