Sharpness of Falconer's estimate in continuous and arithmetic settings, geometric incidence theorems and distribution of lattice points in convex domains

Abstract: In this paper we prove, for all $d \ge 2$, that for no $s<\frac{d+1}{2}$ does $I_s(\mu)<\infty$ imply the canonical Falconer distance problem incidence bound, or the analogous estimate where the Euclidean norm is replaced by the norm generated by a particular convex body $B$ with a smooth boundary and everywhere non-vanishing curvature. Our construction, based on a combinatorial construction due to Pavel Valtr naturally leads us to some interesting connections between the problem under consideration, geometric incidence theorem in the discrete setting and distribution of lattice points in convex domains. We also prove that an example by Mattila can be discretized to produce a set of points and annuli for which the number of incidences is much greater than in the case of the lattice. In particular, we use the known results on the Gauss Circle Problem and a discretized version of Mattila's example to produce a non-lattice set of points and annuli where the number of incidences is much greater than in the case of the standard lattice. Finally, we extend Valtr's example into the setting of vector spaces over finite fields and show that a finite field analog of the key incidence bound is also sharp.