Squares and difference sets in finite fields

Bachoc, C. and Matolcsi, Máté and Ruzsa, Z. Imre (2013) Squares and difference sets in finite fields. INTEGERS, 13. pp. 1-6. ISSN 1553-1732

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Abstract

For infinitely many primes p = 4k+1 we give a slightly improved upper bound for the maximal cardinality of a set B ⊂ Z p such that the difference set B−B contains only quadratic residues. Namely, instead of the ”trivial” bound |B| ≤ √p we prove |B √p | ≤ − 1, under suitable conditions on p. The new bound is valid for approximately three quarters of the primes p = 4k + 1.

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