  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.
