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• W4385281430 endingPage "102264" @default.
• W4385281430 startingPage "102264" @default.
• W4385281430 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∈Fp} and Cp={t3:t∈Fp}, and we use the results on Gauss and Kummer sums. We prove that for any integer k≥3 and for an odd prime number p, the number of k-term arithmetic progressions in Sp is given byp22k+R, where|R|≤(k−24−k−22k−1)⋅p32+ck⋅p 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 ℓ in the set Sp when ℓ∈{3,4,5} and p is an odd prime number. For ℓ=4,5, our formulas are based on the number of points on certain elliptic curves, and the error term is best possible due to the Sato-Tate conjecture." @default.
• W4385281430 created "2023-07-27" @default.
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• W4385281430 date "2023-10-01" @default.
• W4385281430 modified "2023-10-12" @default.
• W4385281430 title "Arithmetic progressions in certain subsets of finite fields" @default.
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• W4385281430 doi "https://doi.org/10.1016/j.ffa.2023.102264" @default.
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