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Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields |
Canad. J. Math. 64(2012), 1036-1057
Published:2011-12-23
Printed: Oct 2012
Printed: Oct 2012
- Doowon Koh,
- Chun-Yen Shen,
Abstract
In this paper we study the extension problem, the
averaging problem, and the generalized Erdős-Falconer distance
problem associated with arbitrary homogeneous varieties in three
dimensional vector spaces over finite fields. In the case when the
varieties do not contain any plane passing through the origin, we
obtain the best possible results on the aforementioned three problems. In
particular, our result on the extension problem modestly generalizes
the result by Mockenhaupt and Tao who studied the particular conical
extension problem. In addition, investigating the Fourier decay on
homogeneous varieties enables us to give complete mapping properties
of averaging operators. Moreover, we improve the size condition on a
set such that the cardinality of its distance set is nontrivial.
| Keywords: | extension problems, averaging operator, finite fields, Erdős-Falconer distance problems, homogeneous polynomial |
| MSC Classifications: |
42B05 - Fourier series and coefficients 11T24 - Other character sums and Gauss sums 52C17 - Packing and covering in $n$ dimensions [See also 05B40, 11H31] |