Toplamsal kombinatorik teorisinden gelen harmonik analiz problemleri

In this thesis, we develop better formulas for the distribution of quadratic residues in certain subsets of finite fields. For this purpose, we consider the sets S_p and C_p, and we use the results on Gauss and Kummer sums. We calculate the number of k-term arithmetic progressions in S_p for any integer k>2 and for an odd prime number p. 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 S_p 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.