On sum of products and the erdos distance problem over finite fields

Abstract

For a prime power Fq, let Fqq be the finite field of q elements. We show that F*qC dA2 for almost every subset A C Fq of cardinality A > q 1/d. Furthermore, if q=p is a prime, and A C Fp of cardinality A > q 1/2 (log p)1/d, then d A2 contains both large and small residues. We also obtain some results of this type for the Erd's distance problem over finite fields. © Australian Mathematical Publishing Association Inc. 2011.