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Recently in joint work with E. Sert, we proved sharp boundedness results ondiscrete fractional integral operators along binary quadratic forms. Presentwork vastly enhances the scope of those results by extending boundedness tobivariate quadratic polynomials. We achieve this in part by establishingconnections to problems on concentration of lattice points on short arcs ofconics, whence we study discrete fractional integrals and lattice pointconcentration from a unified perspective via tools of sieving and diophantineapproximation, and prove theorems that are of interest to researchers in bothsubjects.

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