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

Let ${\mathbb{F}_{q}}$ 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.

Two open problems in the fixed point theory of contractive type mappings on quasimetric spaces

Two open problems in the fixed point theory of quasi metric spaces posed in [Berinde, V. and Choban, M. M., Generalized distances and their associate metrics. Impact on fixed point theory, Creat. Math. Inform., 22 (2013), No. 1, 23–32] are considered. We give a complete answer to the first problem, a partial answer to the second one, and also illustrate the complexity and relevance of these problems by means of four very interesting and comprehensive examples.