2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70353In this paper we prove: If 0 < d < 1, and p is a sufficiently large prime, then if S is a subset of Z/pZ having the least number of three-term arithmetic progressions among all subsets of Z/pZ having at least dp elements, then S has an arithmetic progression of length at least log^{1/4+o(1)} x.Clarified introductionNumber TheoryCombinatorics11P70; 05D99text