SIZE OF DOT PRODUCT SETS DETERMINED BY PAIRS OF SUBSETS OF VECTOR SPACES OVER FINITE FIELDS

In this paper we study the cardinality of the dot prod-uct set generated by two subsets of vector spaces over &macr;nite &macr;elds. 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 Fd q over a &macr;nite &macr;eld Fq with q elements. As a new result, we prove that if E and F are subsets of the parab-oloid and jEjjFj ¸ Cqd for some large C > 1; then j&brvbar;(E; F)j ¸ cq for some 0 < c < 1: In particular, we &macr;nd 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.