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The number of sumsets in a finite field Andrew Granville
 

Summary: The number of sumsets in a finite field
Noga Alon
Andrew Granville
AdriŽan Ubis
Abstract
We prove that there are 2p/2+o(p) distinct sumsets A + B in Fp where |A|, |B|
as p .
1 Introduction
For any subsets A and B of a group G we define the sumset
A + B := {a + b : a A, b B}
There are 2n
subsets of an n element additive group G and every one of them is a sumset,
since A = A+{0} for every A G. However if we restrict our summands to be slightly larger,
then something surprising happens when G = Fp: there are far fewer sumsets:
Theorem 1. Let (x) be any function for which (x) and (x) x/4 as x .
There are exactly 2p/2+o(p)
distinct sumsets in Fp with summands of size (p); that is,
exactly 2p/2+o(p)
distinct sets of the form A + B with |A|, |B| (p) where A, B Fp.
Green and Ruzsa [GrRu] proved that there are only 2p/3+o(p)

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics