Squares and difference sets in finite fields

For infinitely many primes $p=4k+1$ we give a slightly improved upper bound for the maximal cardinality of a set $B\subset {\text{ZZ}}_p$ such that the difference set $B-B$ contains only quadratic residues. Namely, instead of the “trivial” bound $\mathrm{\mid }B\mathrm{\mid }\le \sqrt{p}$ we prove $\mathrm{\mid }B\mathrm{\mid }\le \sqrt{p}-1$, under suitable conditions on $p$. The new bound is valid for approximately three quarters of the primes $p=4k+1$.