Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFalconer type functions in three variables
100
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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 $mathbb{F}_q^d$.
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 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.
For a bilinear form obtained by adding a Dirac mass to a positive definite moment functional in several variables, explicit formulas of orthogonal polynomials are derived from the orthogonal polynomials associated with the moment functional. Explicit formula for the reproducing kernel is also derived and used to establish certain inequalities for classical orthogonal polynomials.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
