Translation invariance in groups of prime order

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⊂Z/pZ satisfies 1<|A|<p/2, then for any positive integer there are at most 2m non-zero elements b∈Z/pZ with |(A+b)?A|?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⊂Z/pZ is an arithmetic progression and m<|A|<p/2, then there are exactly 2m non-zero elements b∈Z/pZ with |(A+b)?A|?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 can be dropped or substantially relaxed.