Incidences between points and generalized spheres over finite fields and related problems

Abstract

AbstractLet ${\mathbb{F}_{q}}$ be a finite field of q elements, where q is a large odd prime power and${Q=a_{1}x_{1}^{c_{1}}+\cdots+a_{d}x_{d}^{c_{d}}\in\mathbb{F}_{q}[x_{1},\ldots,% x_{d}]},$where ${2\leq c_{i}\leq N}$, ${\gcd(c_{i},q)=1}$, and ${a_{i}\in\mathbb{F}_{q}}$ for all ${1\leq i\leq d}$. A Q-sphere is a set of the form ${\bigl{\{}\boldsymbol{x}\in\mathbb{F}_{q}^{d}\mid Q(\boldsymbol{x}-\boldsymbol% {b})=r\bigr{\}}},$where ${\boldsymbol{b}\in\mathbb{F}_{q}^{d},r\in\mathbb{F}_{q}}$. We prove bounds on the number of incidences between a point set ${{{\mathcal{P}}}}$ and a Q-sphere set ${{{\mathcal{S}}}}$, denoted by ${I({{\mathcal{P}}},{{\mathcal{S}}})}$, as the following:$\Biggl{|}I({{\mathcal{P}}},{{\mathcal{S}}})-\frac{|{{\mathcal{P}}}||{{\mathcal% {S}}}|}{q}\Biggr{|}\leq q^{d/2}\sqrt{|{{\mathcal{P}}}||{{\mathcal{S}}}|}.$We also give a version of this estimate over finite cyclic rings ${\mathbb{Z}/q\mathbb{Z}}$, where q is an odd integer. As a consequence of the above bounds, we give an estimate for the pinned distance problem and a bound on the number of incidences between a random point set and a random Q-sphere set in ${\mathbb{F}_{q}^{d}}$. We also study the finite field analogues of some combinatorial geometry problems, namely, the number of generalized isosceles triangles, and the existence of a large subset without repeated generalized distances.