2021/04/21 by Rudnev, Misha, Wheeler, James
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2104.10534
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=λ≠ 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 Möbius 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 L2-moment of the Minkowski distance in H and can therefore fetch the corresponding estimates apropos of the Erdős distinct distance problem.