5 problems

Let $d\ge 2$ and let $A\subseteq \mathbb F_q^d$, where $q$ is an odd prime power. Define … The Erdős–Falconer distance exponent is the smallest $s>0$ such that $|A|\ge Cq^s$ implie…

Let $\mathrm{IM}(2,q)$ denote the maximum size of an induced matching in the point-line incidence graph of $\mathbb{F}_q^2$. Prime-field planar induced matching conjecture. There e…

Let $\mathrm{IM}(d,q)$ denote the maximum size of an induced matching in the point-line incidence graph of $\mathbb{F}_q^d$. High-dimensional induced matching conjecture. For every…

Let $d\ge 2$ be an even integer, let $A,B\subseteq\mathbb F_q^d$, and define … where $\|v\|=\sum_{i=1}^dv_i^2$. Two-set Erdős–Falconer conjecture. If … for a sufficiently large con…

Even-dimensional Erdős–Falconer conjecture. If $|A|\ge Cq^{d/2}$ for a sufficiently large constant $C$ independent of $q$, then