Sharpness of Falconer’s Estimate and the Single Distance Problem in $\mathbb{Z}_{q}^{d}$

Abstract

In the paper introducing the celebrated Falconer distance problem, Falconer proved that the Lebesgue measure of the distance set is positive, provided that the Hausdorff dimension of the underlying set is greater than $\frac{d+1} {2}$. His result is based on the estimate

$$\displaystyle{ \mu \times \mu \{(x,y): 1 \leq \vert x - y\vert \leq 1+\varepsilon \} \lesssim \varepsilon, }$$

(1)

where μ is a Borel measure satisfying the energy estimate $I_{s}(\mu ) =\int \int \vert x - y\vert ^{-s}$ d μ(x)d μ(y) < ∞ for $s > \frac{d+1} {2}$. An example due to Mattila ([15], Remark 4.5; [14]) shows in two dimensions that for no $s < \frac{3} {2}$ does I _s (μ) < ∞ imply (1). His construction can be extended to three dimensions, but not to dimensions four and higher. Mattila’s example, as well as Falconer’s result, readily applies to the case when the Euclidean norm in (1) is replaced by a norm generated by a convex body with a smooth boundary and nonvanishing Gaussian curvature.

In this paper we prove, for all d ≥ 2, that for no $s < \frac{d+1} {2}$ does I _s (μ) < ∞ imply (1) 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 nonvanishing curvature. We also study the analog of the single distance problem in vector spaces over $\mathbb{Z}_{q}$, the integers modulo q, and obtain a new geometric incidence result. Our constructions involve extending a two-dimensional combinatorial construction due to Valtr [20] who previously used to establish sharpness of some classical results in geometric combinatorics.

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