## Abstract

Let ${\mathrm{\u0161\x9d\x94\xbd}}_{q}$ be a finite field of *q* elements, where *q* is a large odd prime power and

$Q={a}_{1}\u0101\x81\xa2{x}_{1}^{{c}_{1}}+\mathrm{\u0101\x8b\AE}+{a}_{d}\u0101\x81\xa2{x}_{d}^{{c}_{d}}\u0101\x88\x88{\mathrm{\u0161\x9d\x94\xbd}}_{q}\u0101\x81\xa2[{x}_{1},\mathrm{\u0101\x80\xa6},{x}_{d}],$

where $2\u0101\x89\xa4{c}_{i}\u0101\x89\xa4N$, $\mathrm{gcd}\u0101\x81\u201d({c}_{i},q)=1$, and ${a}_{i}\u0101\x88\x88{\mathrm{\u0161\x9d\x94\xbd}}_{q}$ for all $1\u0101\x89\xa4i\u0101\x89\xa4d$. A *Q*
*-sphere* is a set of the form

$\{\mathrm{\u0161\x9d\x92\x99}\u0101\x88\x88{\mathrm{\u0161\x9d\x94\xbd}}_{q}^{d}\u0101\x88\pounds Q\u0101\x81\xa2(\mathrm{\u0161\x9d\x92\x99}-\mathrm{\u0161\x9d\x92\x83})=r\},$

where $\mathrm{\u0161\x9d\x92\x83}\u0101\x88\x88{\mathrm{\u0161\x9d\x94\xbd}}_{q}^{d},r\u0101\x88\x88{\mathrm{\u0161\x9d\x94\xbd}}_{q}$. We prove bounds on the number of incidences between a point set $\mathcal{\u0161\x9d\x92\xab}$ and a *Q*-sphere set $\mathcal{\u0161\x9d\x92\circledR}$, denoted by $I\u0101\x81\xa2(\mathcal{\u0161\x9d\x92\xab},\mathcal{\u0161\x9d\x92\circledR})$, as the following:

$\left|I\u0101\x81\xa2(\mathcal{\u0161\x9d\x92\xab},\mathcal{\u0161\x9d\x92\circledR})-\frac{|\mathcal{\u0161\x9d\x92\xab}|\u0101\x81\xa2|\mathcal{\u0161\x9d\x92\circledR}|}{q}\right|\u0101\x89\xa4{q}^{d/2}\u0101\x81\xa2\sqrt{|\mathcal{\u0161\x9d\x92\xab}|\u0101\x81\xa2|\mathcal{\u0161\x9d\x92\circledR}|}.$

We also give a version of this estimate over finite cyclic rings $\mathrm{\u0101\x84\xa4}/q\u0101\x81\xa2\mathrm{\u0101\x84\xa4}$, 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 ${\mathrm{\u0161\x9d\x94\xbd}}_{q}^{d}$. 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.

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