  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.
