On subsets of Fqn containing no kk-term progressions

In this paper we prove that for any fixed integer kk and any prime power q≥kq≥k, there exists a subset of Fq2k of size q2(k−1)+qk−1−1q2(k−1)+qk−1−1 which contains no kk points on a line, and hence no kk-term arithmetic progressions. As a corollary we obtain an asymptotic lower bound as n→∞n→∞ for rk(Fqn) when q≥kq≥k, which can be interpreted as the finite field analogue of Behrend’s construction for longer progressions.