Translation invariance in groups of prime order

Abstract

We prove that there is an absolute constant $c>0$ with the following property: if $Z/pZ$ denotes the group of prime order $p$, and a subset $A\subset Z/pZ$ satisfies $1<|A|<p/2$, then for any positive integer $m<\min \{c|A|/\ln |A|,\sqrt{p/8}\}$ there are at most $2m$ non-zero elements $b\in Z/pZ$ with $|(A+b)\setminus A|\le m$. This (partially) extends onto prime-order groups the result, established earlier by S. Konyagin and the present author for the group of integers. We notice that if $A\subset Z/pZ$ is an arithmetic progression and $m<|A|<p/2$, then there are exactly $2m$ non-zero elements $b\in Z/pZ$ with $|(A+b)\setminus A|\le m$. Furthermore, the bound $c|A|/\ln |A|$ is best possible up to the value of the constant $c$. On the other hand, it is likely that the assumption $m<\sqrt{p/8}$ can be dropped or substantially relaxed.