Incidence bounds with M\"obius hyperbolae in positive characteristic

We prove new incidence bounds between a plane point set, which is a Cartesian product, and a set of translates $H$ of the hyperbola $xy=\lambda \mathrm{\ne }0$, over a field of asymptotically large positive characteristic $p$. They improve recent bounds by Shkredov, which are based on using explicit incidence estimates in the early terminated procedure of repeated applications of the Cauchy-Schwarz inequality, underlying many qualitative results related to growth and expansion in groups. The improvement – both quantitative, plus we are able to deal with a general $H$, rather than a Cartesian product – is mostly due to a non-trivial “intermediate” bound on the number of $k$-rich M"obius hyperbolae in positive characteristic. In addition, we make an observation that a certain energy-type quantity in the context of $H$ can be bounded via the $L^2$-moment of the Minkowski distance in $H$ and can therefore fetch the corresponding estimates apropos of the Erd\H{o}s distinct distance problem.