Size of dot product sets determined by pairs of subsets of vector spaces over finite fields

Abstract

In this paper we study the cardinality of the dot product set generated by two subsets of vector spaces over finite fields. We notice that the results on the dot product problems for one set can be simply extended to two sets. Let E and F be subsets of the d-dimensional vector space F_q^d over a finite field F_q with q elements. As a new result, we prove that if E and F are subsets of the paraboloid and |E||F|\geq C q^d for some large C>1, then |\Pi(E,F)|\geq c q for some 0<c<1. In particular, we find a connection between the size of the dot product set and the number of lines through both the origin and a nonzero point in the given set E. As an application of this observation, we obtain more sharpened results on the generalized dot product set problems. The discrete Fourier analysis and geometrical observation play a crucial role in proving our results.