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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 - Scopus: 1
Level Set Estimates for the Discrete Frequency Function
(Springer Verlag, 2019) Temur, Faruk
We introduce the discrete frequency function as a possible new approach to understanding the discrete Hardy-Littlewood maximal function. Considering that the discrete Hardy-Littlewood maximal function is given at each integer by the supremum of averages over intervals of integer length, we define the discrete frequency function at that integer as the value at which the supremum is attained. After verifying that the function is well-defined, we investigate size and smoothness properties of this function.
The Frequency Function and Its Connections To the Lebesgue Points and the Hardy-Littlewood Maximal Function
(TÜBİTAK, 2019) Temur, Faruk
The 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.
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.
A Quantitative Balian-Low Theorem for Higher Dimensions
(De Gruyter, 2020) Temur, Faruk
We extend the quantitative Balian-Low theorem of Nitzan and Olsen to higher dimensions. We use Zak transform methods and dimension reduction. The characterization of the Gabor-Riesz bases by the Zak transform allows us to reduce the problem to the quasiperiodicity and the boundedness from below of the Zak transforms of the Gabor-Riesz basis generators, two properties for which dimension reduction is possible. © 2018 Walter de Gruyter GmbH, Berlin/Boston 2018.

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