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Register a new userA Polynomial Roth Theorem for Corners in Finite Fields
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We prove a Roth type theorem for polynomial corners in the finite field setting. Let $phi_1$ and $phi_2$ be two polynomials of distinct degree. For sufficiently large primes $p$, any subset $ A subset mathbb F_p times mathbb F_p$ with $ lvert Arvert > p ^{2 - frac1{16}} $ contains three points $ (x_1, x_2) , (x_1 + phi_1 (y), x_2), (x_1, x_2 + phi_2 (y))$. The study of these questions on $ mathbb F_p$ was started by Bourgain and Chang. Our Theorem adapts the argument of Dong, Li and Sawin, in particular relying upon deep Weil type inequalities established by N. Katz.
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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 $mathbb{F}_q^d$.
Let $phi(x, y)colon mathbb{R}^dtimes mathbb{R}^dto mathbb{R}$ be a function. We say $phi$ is a Mattila--Sj{o}lin type function of index $gamma$ if $gamma$ is the smallest number satisfying the property that for any compact set $Esubset mathbb{R}^d$, $phi(E, E)$ has a non-empty interior whenever $dim_H(E)>gamma$. The usual distance function, $phi(x, y)=|x-y|$, is conjectured to be a Mattila--Sj{o}lin type function of index $frac{d}{2}$. In the setting of finite fields $mathbb{F}_q$, this definition is equivalent to the statement that $phi(E, E)=mathbb{F}_q$ whenever $|E|gg q^{gamma}$. The main purpose of this paper is to prove the existence of such functions with index $frac{d}{2}$ in the vector space $mathbb{F}_q^d$.
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 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.
Let $fin mathbb{R}[x, y, z]$ be a quadratic polynomial that depends on each variable and that does not have the form $g(h(x)+k(y)+l(z))$. Let $A, B, C$ be compact sets in $mathbb{R}$. Suppose that $dim_H(A)+dim_H(B)+dim_H(C)>2$, then we prove that the image set $f(A, B, C)$ is of positive Lebesgue measure. Our proof is based on a result due to Eswarathasan, Iosevich, and Taylor (Advances in Mathematics, 2011), and a combinatorial argument from the finite field model.
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