Extension theorems for the Fourier transform associated with non-degenerate quadratic surfaces in vector spaces over finite fields

Details Extension theorems for the Fourier transform associated with non-degenerate quadratic surfaces in vector spaces over finite fields Extension theorems for the Fourier transform associated with non-degenerate quadratic surfaces in vector spaces over finite fields

2008-04-28

arxiv.org/abs/0804.4505v1

Mathematics

Classical Analysis and ODEs

18 pages, 4 figures

Scientific paper

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.

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