turkmath.org

The study of discrete fractional integral operators began with an article of Arkhipov and Oskolkov on boundedness of certain multipliers, and for over thirty years concentrated on cases with translation invariant or quasitranslation invariant phase polynomial, as these cases are amenable to the Fourier transform and the Hardy-Littlewood circle method. Recently in joint work with E. Sert, we introduced methods from arithmetic to study discrete fractional integral operators along quadratic bivariate polynomials in their full generality. This is achieved by combining the information regarding distribution of lattice points on conics gleaned via classical theory of binary quadratic forms and sieving with very careful partitioning of sums under question. This effort as a side benefit yields new results on various conjectures on concentration of lattice points on conics and makes connections to diophantine approximation. 

In this talk we will give a brief summary of these results on fractional integrals and lattice points, together with their mathematical context and main ideas of their proofs.