Sharp extension theorems and Falconer distance problems for algebraic curves in two dimensional vector spaces over finite fields

2010/03/22 by Doowon Koh, Chun-Yen Shen, Koh, Doowon +2 · 2 citations
Mathematics · #Advanced Harmonic Analysis Research #Analytic Number Theory Research #Mathematical Approximation and Integration #math.CA

paper · pdf · doi:10.48550/arxiv.1003.4240

18 pages. Revised version incorporates the referee's report

arxiv created 2010/11/30 · arxiv updated 2010/12/01

Abstract

In this paper we study extension theorems associated with general varieties in two dimensional vector spaces over finite fields. Applying Bezout's theorem, we obtain the sufficient and necessary conditions on general curves where sharp L^p-L^r extension estimates hold. Our main result can be considered as a nice generalization of works by Mochenhaupt and Tao and Iosevich and Koh. As an application of our sharp extension estimates, we also study the Falconer distance problems in two dimensions.

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• Sharp endpoint extension inequalities for the moment curve on finite fields
• The Erdős-Falconer distance problem between arbitrary sets and k-coordinatable sets in finite fields

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