Discrete fractional integrals, lattice points on short arcs, and diophantine approximation

TL;DR

This paper extends sharp boundedness results of discrete fractional integral operators from binary quadratic forms to bivariate quadratic polynomials, connecting lattice point concentration on conics with diophantine approximation.

Contribution

It significantly broadens the scope of previous results by establishing boundedness for more general quadratic polynomials and linking lattice point problems with diophantine approximation techniques.

Findings

Boundedness results extended to bivariate quadratic polynomials

Connections established between lattice point concentration and diophantine approximation

Unified approach using sieving and diophantine tools

Abstract

Recently in joint work with E. Sert, we proved sharp boundedness results on discrete fractional integral operators along binary quadratic forms. Present work vastly enhances the scope of those results by extending boundedness to bivariate quadratic polynomials. We achieve this in part by establishing connections to problems on concentration of lattice points on short arcs of conics, whence we study discrete fractional integrals and lattice point concentration from a unified perspective via tools of sieving and diophantine approximation, and prove theorems that are of interest to researchers in both subjects.