ON SUM OF PRODUCTS AND THE ERDŐS DISTANCE PROBLEM OVER FINITE FIELDS

Abstract

AbstractFor a prime powerq, let 𝔽qbe the finite field ofqelements. We show that 𝔽*q⊆d𝒜2for almost every subset 𝒜⊂𝔽qof cardinality ∣𝒜∣≫q1/d. Furthermore, ifq=pis a prime, and 𝒜⊆𝔽pof cardinality ∣𝒜∣≫p1/2(logp)1/d, thend𝒜2contains both large and small residues. We also obtain some results of this type for the Erdős distance problem over finite fields.