In this paper (joint with Imre Ruzsa) we count the number of subsets of Zp of various types. Let |SS(Zp)| be the number of sets having the form A + A, where A is a subset of Zp, and let |SF(Zp)| denote the number of sumfree subsets of Zp (that is to say sets A such that A3 contains no solutions to a1 + a2 = a3). We show that both |SS(Zp)| and |SF(Zp)| are bounded above by 2p/3 + o(p).
The quantity |SS(Zp)| does not seem to have been considered before but |SF(Zp)| has been studied by several authors. In particular it was recently by Lev and Schoen that |SF(Zp)| < 20.498p.
Our bounds are tight up to the o(p) in the exponent. Indeed Lev and Schoen showed that |SF(Zp)| >> p2p/3, and in the paper we prove that |SS(Zp)| >> p22p/3.
Generalising our results to more general abelian groups is not at all straightforward, and will be the subject of a second paper.
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