74 research outputs found
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 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.Comment: 11 page
Extension theorems for the Fourier transform associated with non-degenerate quadratic surfaces in vector spaces over finite fields
We study the restriction of the Fourier transform to quadratic surfaces in
vector spaces over finite fields. In two dimensions, we obtain the sharp result
by considering the sums of arbitrary two elements in the subset of quadratic
surfaces on two dimensional vector spaces over finite fields. For higher
dimensions, we estimate the decay of the Fourier transform of the
characteristic functions on quadratic surfaces so that we obtain the
Tomas-Stein exponent. Using incidence theorems, we also study the extension
theorems in the restricted settings to sizes of sets in quadratic surfaces.
Estimates for Gauss and Kloosterman sums and their variants play an important
role.Comment: 18 pages, 4 figure
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