  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 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
$\mathbb{R}^d$.
