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

Abstract

Abstract Let 𝔽 q ${\mathbb{F}_{q}}$ be a finite field of q elements, where q is a large odd prime power and Q = a 1 ⁢ x 1 c 1 + ⋯ + a d ⁢ x d c d ∈ 𝔽 q ⁢ [ x 1 , … , x d ] , ${Q=a_{1}x_{1}^{c_{1}}+\cdots+a_{d}x_{d}^{c_{d}}\in\mathbb{F}_{q}[x_{1},\ldots,% x_{d}]},$ where 2 ≤ c i ≤ N ${2\leq c_{i}\leq N}$ , gcd ⁡ ( c i , q ) = 1 ${\gcd(c_{i},q)=1}$ , and a i ∈ 𝔽 q ${a_{i}\in\mathbb{F}_{q}}$ for all 1 ≤ i ≤ d ${1\leq i\leq d}$ . A Q -sphere is a set of the form { 𝒙 ∈ 𝔽 q d ∣ Q ⁢ ( 𝒙 - 𝒃 ) = r } , ${\bigl{\{}\boldsymbol{x}\in\mathbb{F}_{q}^{d}\mid Q(\boldsymbol{x}-\boldsymbol% {b})=r\bigr{\}}},$ where 𝒃 ∈ 𝔽 q d , r ∈ 𝔽 q ${\boldsymbol{b}\in\mathbb{F}_{q}^{d},r\in\mathbb{F}_{q}}$ . We prove bounds on the number of incidences between a point set 𝒫 ${{{\mathcal{P}}}}$ and a Q-sphere set 𝒮 ${{{\mathcal{S}}}}$ , denoted by I ⁢ ( 𝒫 , 𝒮 ) ${I({{\mathcal{P}}},{{\mathcal{S}}})}$ , as the following: | I ⁢ ( 𝒫 , 𝒮 ) - | 𝒫 | ⁢ | 𝒮 | q | ≤ q d / 2 ⁢ | 𝒫 | ⁢ | 𝒮 | . $\Biggl{|}I({{\mathcal{P}}},{{\mathcal{S}}})-\frac{|{{\mathcal{P}}}||{{\mathcal% {S}}}|}{q}\Biggr{|}\leq q^{d/2}\sqrt{|{{\mathcal{P}}}||{{\mathcal{S}}}|}.$ We also give a version of this estimate over finite cyclic rings ℤ / q ⁢ ℤ ${\mathbb{Z}/q\mathbb{Z}}$ , 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 𝔽 q d ${\mathbb{F}_{q}^{d}}$ . 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.