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 prod- uct set generated by two subsets of vector spaces over flnite flelds. 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 spaceF d over a flnite fleldFq with q elements. As a new result, we prove that if E and F are subsets of the parab- oloid and jEjjFj ‚ Cq d for some large C > 1; then jƒ(E;F)j ‚ cq for some 0 < c < 1: In particular, we flnd 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 gener- alized dot product set problems. The discrete Fourier analysis and geometrical observation play a crucial role in proving our results.