Long Arithmetic Progressions in Sum-Sets and the Number x-Sum-Free Sets

In this paper we obtain optimal bounds for the length of thelongest arithmetic progression in various kinds of sum-sets.As an application, we derive a sharp estimate for the numberof sets A of residues modulo a prime n such that no subsum ofA equals x modulo n, where x is a fixed residue modulo n. 2000Mathematics Subject Classification 05A16, 11B25, 11P32.