Geometric progressions in distance sets in odd dimensions over prime fields

Abstract

For $E\subset\mathbb{F}_p^d$, let $Δ(E)=\{Q_d(x-y):x,y\in E\}$, where $Q_d(z)=z_1^2+\cdots+z_d^2$. We introduce and study the problem of the existence of nondegenerate three-term geometric progressions in the distance set $Δ(E)$ when $d$ is odd. If $M_{GP,d}(p)$ denotes the largest size of a set whose distance set contains no such progression, then, for every prime $p\geq7$ and every integer $m\geq0$, \[ p^mM_{GP,1}(p)\leq M_{GP,2m+1}(p) \leq4p^mM_{GP,1}(p). \] For all sufficiently large primes $p$, we also prove \[ p^{m+\frac{7}{26}}\leq M_{GP,2m+1}(p) \leq Cp^{m+\frac{16}{25}}, \] where $C$ is an absolute positive constant.