An algorithm for finding a nearly minimal balanced set in 𝔽_{𝕡}

Abstract

For a prime p p , we call a non-empty subset S S of the group F p \mathbb {F}_p balanced if every element of S S is the midterm of a three-term arithmetic progression, contained in S S . A result of Browkin, Diviš and Schinzel implies that the size of a balanced subset of F p \mathbb {F}_p is at least log 2 ⁡ p + 1 \log _{2} p + 1 . In this paper we present an efficient algorithm which yields a balanced set of size ( 1 + o ( 1 ) ) log 2 ⁡ p (1 + o(1)) \log _{2} p as p p grows.