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Register a new userSharp extension theorems and Falconer distance problems for algebraic curves in two dimensional vector spaces over finite fields
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In this paper we study extension theorems associated with general varieties in two dimensional vector spaces over finite fields. Applying Bezouts 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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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 a new $L^2to L^r$ extension estimate for paraboloids in dimensions $d=4k+3$ and $qequiv 3mod 4$, which improves significantly the recent exponent obtained by the first listed author. In the case of spheres, we introduce a way of using textit{the first association scheme graph} to analyze energy sets, and as a consequence, we obtain new $L^pto L^4$ extension theorems for spheres of primitive radii in odd dimensions, which break the Stein-Tomas result toward $L^pto L^4$ 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 $L^pto L^4$ estimates for spheres with primitive radii are much stronger than those for paraboloids. Based on new estimates, we will also clarify conjectures on finite field extension problem for spheres. This results in a reasonably complete description of finite field extension theorems for spheres. The second purpose is to show that there is a connection between the restriction conjecture associated to paraboloids and the ErdH{o}s-Falconer distance conjecture over finite fields. The last is to prove that the ErdH{o}s-Falconer distance conjecture holds in odd-dimensional spaces when we study distances between two sets: one set lies on a variety (paraboloids or spheres), and the other set is arbitrary in $mathbb{F}_q^d$.
We prove a point-wise and average bound for the number of incidences between points and hyper-planes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering ${mathbb F}_q$, the finite field with q elements, by $A cdot A+... +A cdot A$, where A is a subset ${mathbb F}_q$ of sufficiently large size. We also use the incidence machinery we develope and arithmetic constructions to study the Erdos-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdos-Falconer distance conjecture does not hold in this setting due to the influence of the arithmetic. On the positive side, we obtain good exponents for the Erdos -Falconer distance problem for subsets of the unit sphere in $mathbb F_q^d$ and discuss their sharpness. This results in a reasonably complete description of the Erdos-Falconer distance problem in higher dimensional vector spaces over general finite fields.
We study the finite field extension estimates for Hamming varieties $H_j, jin mathbb F_q^*,$ defined by $H_j={xin mathbb F_q^d: prod_{k=1}^d x_k=j},$ where $mathbb F_q^d$ denotes the $d$-dimensional vector space over a finite field $mathbb F_q$ with $q$ elements. We show that although the maximal Fourier decay bound on $H_j$ away from the origin is not good, the Stein-Tomas $L^2to L^r$ extension estimate for $H_j$ holds.
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.
We prove that a sufficiently large subset of the $d$-dimensional vector space over a finite field with $q$ elements, $ {Bbb F}_q^d$, contains a copy of every $k$-simplex. Fourier analytic methods, Kloosterman sums, and bootstrapping play an important role.
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