TR Dizin İndeksli Yayınlar / TR Dizin Indexed Publications Collection
Permanent URI for this collectionhttps://hdl.handle.net/11147/7149
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Article Discrete Fractional Integrals, Lattice Points on Short Arcs, and Diophantine Approximation(TÜBİTAK - Türkiye Bilimsel ve Teknolojik Araştırma Kurumu, 2022) Temur, FarukRecently 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.Article The Frequency Function and Its Connections To the Lebesgue Points and the Hardy-Littlewood Maximal Function(TÜBİTAK, 2019) Temur, FarukThe aim of this work is to extend the recent work of the author on the discrete frequency function to the more delicate continuous frequency function tau, and further to investigate its relations to the Hardy-Littlewood maximal function M, and to the Lebesgue points. We surmount the intricate issue of measurability of tau f by approaching it with a sequence of carefully constructed auxiliary functions for which measurability is easier to prove. After this, we give analogues of the recent results on the discrete frequency function. We then connect the points of discontinuity of Mf for f simple to the zeros of tau f, and to the non-Lebesgue points of f.