Please use this identifier to cite or link to this item:
https://hdl.handle.net/11147/13781
Title: | Arithmetic progressions in certain subsets of finite fields | Authors: | Eyidoğan, Sadık Göral, Haydar Kutlu, Mustafa Kutay |
Keywords: | Arithmetic progressions Szemeredi's theorem Arithmetic geometry Weil estimates Sato-Tate conjecture |
Publisher: | Elsevier | Abstract: | In this note, we focus on how many arithmetic progressions we have in certain subsets of finite fields. For this purpose, we consider the sets Sp = {t2 : t & ISIN; Fp} and Cp = {t3 : t & ISIN; Fp}, and we use the results on Gauss and Kummer sums. We prove that for any integer k & GE; 3 and for an odd prime number p, the number of k-term arithmetic progressions in Sp is given by p2 2k + R, where and ck is a computable constant depending only on k. The proof also uses finite Fourier analysis and certain types of Weil estimates. Also, we obtain some formulas that give the exact number of arithmetic progressions of length in the set Sp when & ISIN; {3,4, 5} and p is an odd prime number. For = 4, 5, our formulas are based on the number of points on | URI: | https://doi.org/10.1016/j.ffa.2023.102264 https://hdl.handle.net/11147/13781 |
ISSN: | 1071-5797 1090-2465 |
Appears in Collections: | Mathematics / Matematik Scopus İndeksli Yayınlar Koleksiyonu / Scopus Indexed Publications Collection WoS İndeksli Yayınlar Koleksiyonu / WoS Indexed Publications Collection |
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1-s2.0-S1071579723001065-main.pdf | 694.56 kB | Adobe PDF | View/Open |
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