Falconer-type estimates for dot products

We present a family of sharpness examples for Falconer-type single dot product results. In particular, for $d\geq 2,$ for any $s<\frac{d+1}{2},$ we construct a Borel probability measure $\mu$ satisfying the energy estimate $I_s(\mu)<\infty,$ yet the estimate \begin{equation} (\mu \times \mu)\{(x,y):1\leq x\cdot y \leq 1+\epsilon\} \leq C\epsilon \end{equation} does not hold with constants independent of $\epsilon$. It is known (\cite{EIT11}) that such an estimate always holds with $C$ independent of $\epsilon$ if $I_{\frac{d+1}{2}}(\mu)<\infty$. Thus our estimate proves the sharpness of the dimensional threshold in this result and generalizes similar results (\cite{Mat95}, \cite{IS16}) established in the case when the dot product $x \cdot y$ is replaced by the Euclidean distance function $|x-y|$, or, more generally, ${||x-y||}_K$, the distance that comes from the norm induced by a symmetric convex body $K$ with a smooth boundary and non-vanishing curvature. Our constructions are partially based on ideas that come from discrete incidence theory.