turkmath.org

In 1975, Szemeredi gave an affirmative answer to Erdös and Turan's conjecture which states that any subset of positive integers with a positive upper density contains arbitrarily long arithmetic progressions. Szemeredi-type problems have also been extensively studied in subsets of finite fields. While much work has been done on the problem of whether subsets of finite fields contain arithmetic progressions, in this talk we concentrate on how many arithmetic progressions we have in certain subsets of finite fields. The technique is based on certain types of Weil estimates. We obtain an asymptotic for the number of k-term arithmetic progressions in squares with a better error term. Moreover our error term is sharp and best possible when k is small, owing to the Sato-Tate conjecture. This work is supported by the Scientific and Technological Research Council of Turkey with the project number 122F027.