turkmath.org

 Arkhipov and Oskolkov initiated the study of discrete fractional integrals with their work on boundedness of certain Fourier multipliers about thirty years ago. Since then study of these operators, carried out mostly by E. Stein, his students and collaborators, concentrated on cases with translation invariant or quasitranslation invariant phase polynomial, to exploit applicability of  the Fourier transform and the Hardy-Littlewood circle method to these cases. In 2018, in joint work with E. Sert, we introduced methods from number theory to study discrete fractional integral operators along binary quadratic forms. More specifically we investigate the distribution of lattice points  on conics via classical theory of binary quadratic forms  and apply these to discrete fractional integral operators using some delicate decompositions. In 2021, in continuation of this work, using further information about  distribution of lattice points  on conics obtained from sieving,   analogous results for the even more general case of bivariate quadratic polynomials were obtained. Also in this latter work, connections to diophantine approximation were established, and new results on certain well-known conjectures on lattice points concentration were obtained. In this talk we will review these developments, together with their historical and mathematical context.