Akademik Veri Yönetim Sistemi
FINITE FIELDS AND THEIR APPLICATIONS, cilt.91, ss.1-59, 2023 (SCI-Expanded)
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 $S_p =\{t^2: t\in \mathbb{F}_{p} \}$ and $C_{p} = \left\lbrace t^{3} : \ t \in \mathbb{F}_{p}\right\rbrace $, 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 $ S_{p} $ is given by $$ \frac{p^{2}}{2^{k}}+ R,$$ where $$ \lvert R \rvert \le \bigg(\frac{k-2}{4}-\frac{k-2}{2^{k-1}}\bigg) \cdot p^{\frac{3}{2}} + c_{k} \cdot p $$ and $ c_{k} $ 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 $ \ell $ in the set $ S_{p} $ when $ \ell \in \left\lbrace 3, 4, 5\right\rbrace $ and $ p $ is an odd prime number. For $\ell=4,5$, 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.