Additive structures imply more distances in $\mathbb{F}_q^d$
Daewoong Cheong, Gennian Ge, Doowon Koh, Thang Pham, Dung The Tran, Tao Zhang
TL;DR
The paper advances the finite-field Erdős–Falconer distance problem by exploiting Fraser's $(u,s)$--Salem framework with a focus on the even case $u=4$. Through a precise identity linking $\| abla E\|_4$ to the fourth additive energy $\Lambda_4(E)$, it derives two complementary incidence-based bounds that yield improved distance-threshold exponents $\alpha(d,4,s) \le \min\{\frac{d+2}{4s+1},\frac{d+4}{8s}\}$ for $s\in[\tfrac14,\tfrac12]$, with several refinements for spheres, subgroups, and varieties. The work also provides sharp point–hyperplane incidence bounds for Salem sets, two-set distance results, and a unified conjectural framework that clarifies the spherical-distance regime, including odd-dimensional behavior. Concrete constructions show near-sharpness of the proposed thresholds, while applications to multiplicative subgroups and algebraic varieties demonstrate broad impact. The paper thus advances both the methodology and the range of settings where positive-proportion distance results can be guaranteed in finite fields.
Abstract
For a set $E \subseteq \mathbb{F}_q^d$, the distance set is defined as $Δ(E) := \{\|\mathbf{x} - \mathbf{y}\| : \mathbf{x}, \mathbf{y} \in E\}$, where $\|\cdot\|$ denotes the standard quadratic form. We investigate the Erdős--Falconer distance problem within the flexible class of $(u, s)$--Salem sets introduced by Jonathan M. Fraser, with emphasis on the even case $u = 4$. By exploiting the exact identity between $\|\widehat{E}\|_4$ and the fourth additive energy $Λ_4(E)$, we prove that quantitative gains in $Λ_4(E)$ force the existence of many distances. In particular, for a $(4, s)$--Salem set $E\subset \mathbb{F}_q^d$ with $d \geq 2$, if \[ |E|\gg q^{\min\left\{\frac{d+2}{4s+1}, \frac{d+4}{8s}\right\}}, \] then $E$ determines a positive proportion of all distances. This strictly improves Fraser's threshold of $\frac{d}{4s}$ and the Iosevich-Rudnev bound of $q^{\frac{d+1}{2}}$ in certain parameter ranges. As applications, we obtain improved thresholds for multiplicative subgroups and sets on arbitrary varieties, and establish a sharp incidence bound for Salem sets that is of independent interest in incidence geometry. Moreover, our methods give sharp lower bounds for the number of distinct distances determined by two different sets. We also propose a unified conjecture for $(4, s)$--Salem sets that reconciles known bounds and pinpoints the odd-dimensional sphere regime: in odd dimensions $d \geq 3$, the often-cited $\frac{d-1}{2}$ threshold does not follow without additional structures. This provides a clear picture of the spherical distance conjecture.