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 $\|x-y\| \neq R_n$ for all $x,y\in A$ and $\mu (A\cap B_{R_n})\geq f(R_n)\mu (B_{R_n})$ for all $n\in \mathbb {N}$, where $B_R$ is the ball of radius $R$ centered at the origin and $\mu$ is a 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 $\mathbb {R}^d$.