Extension theorems and a connection to the Erdős-Falconer distance problem over finite fields
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Date
2021-10-15
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Journal Article
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yes
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Abstract
The first purpose of this paper is to provide new finite field extension theorems for paraboloids and spheres. By using the unusual good Fourier transform of the zero sphere in some specific dimensions, which has been discovered recently in the work of Iosevich, Lee, Shen, and the first and second listed authors (2018), we provide new L2→Lr extension estimates for paraboloids in certain odd dimensions with −1 non-square, which improves significantly the recent exponent obtained by the first listed author. In the case of spheres, we introduce a way of using the first association scheme graph to analyze energy sets, and as a consequence, we obtain new Lp→L4 extension theorems for spheres of primitive radii in odd dimensions, which break the Stein-Tomas result toward Lp→L4 which has stood for more than ten years. Most significantly, it follows from the results for spheres that there exists a different extension phenomenon between spheres and paraboloids in odd dimensions, namely, the Lp→L4 estimates for spheres with primitive radii are much stronger than those for paraboloids. The second purpose is to show that there is a connection between the restriction conjecture associated to paraboloids and the Erdős-Falconer distance conjecture over finite fields. The last is to prove that the Erdős-Falconer distance conjecture holds in odd dimensional spaces when we study distances between two sets: one set lies on a variety (a paraboloid or a sphere), and the other set is arbitrary in vector spaces over finite fields.
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published
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Volume
281 (8)
Pages / Article No.
109137
Publisher
Elsevier
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Subject
Extension theorems; Finite fields; Erdős-Falconer distance problem; Energy bounds
Organisational unit
03457 - Welzl, Emo (emeritus) / Welzl, Emo (emeritus)
02643 - Institut für Theoretische Informatik / Inst. Theoretical Computer Science
Notes
Funding
191067 - Erdos-Falconer Distance Conjecture and Related Topics (SNF)