dimension of $\Delta_{x, K}(E)$ is $\ge \frac{4s}{3}-\frac{2}{3}$. Keleti and Shmerkin ([@KS18]) obtained the lower bound $\frac{1}{4}(1+s+\sqrt{3s(2-s)})$ in the case of the Euclidean metric. Their estimate is better than ours near $s=1$, but ours is preferable as $s$ nears $\frac{5}{4}$. The sketch of this argument is given in Appendix where we also discuss the complications of replacing the upper Minkowski dimension by the Hausdorff dimension in the claim above.

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Falconer’s distance problem can be thought of as a continuous analogue of a combinatorial problem raised by Erdős in [@Erd45]: given a set $P$ of $N$ points in ${\mathbb{R}}^d$, what is the smallest possible cardinality of $\Delta(P)$. A grid is the best known example in all dimensions. In two dimensions, Guth and Katz [@GK15] proved a lower bound for $|\Delta(P)|$ which nearly matches the grid example (up to a factor of $\log^{1/2} N$). In higher dimensions, there is a larger gap, and the best known result is due to Solymosi and Vu [@SV08]. The Erdős distinct distance problem also makes sense for general norms and much less is known about it. In the planar case, if $K$ is smooth and has strictly positive curvature, the best known bound says that if 