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Discrete Fractional Integrals, Lattice Points on Short Arcs, and Diophantine Approximation
(TÜBİTAK - Türkiye Bilimsel ve Teknolojik Araştırma Kurumu, 2022) Temur, Faruk
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.
Citation - WoS: 1
Citation - Scopus: 1
Discrete Fractional Integral Operators With Binary Quadratic Forms as Phase Polynomials
(Academic Press, 2019) Temur, Faruk; Sert, Ezgi
We give estimates on discrete fractional integral operators along binary quadratic forms. These operators have been studied for 30 years starting with the investigations of Arkhipov and Oskolkov, but efforts have concentrated on cases where the phase polynomial is translation invariant or quasi-translation invariant. This work presents the first results for operators with neither translation invariant nor quasi-translation invariant phase polynomials. (C) 2019 Elsevier Inc. All rights reserved.

Mathematics / Matematik