Related papers: Density criteria for Fourier uniqueness phenomena …
We use weighted polynomial approximation to prove the existence of a compact set K with non-empty interior and a function f is dense in the space A(K) of all continuous functions on K that are holomorphic in the interior of K, endowed with…
In order to extend the study of uniqueness property of multi-dimensional systems of stochastic differential equations, in this paper, we look at the following three-dimensional system of equations, of which the two-dimensional case was…
Using the notion of higher-order Fourier dimension introduced in \cite{M2} (which was a sort of psuedorandomness condition stemming from the Gowers norms of Additive Combinatorics), we prove a maximal theorem and corresponding…
By making use of arithmetic information inequalities, we give a strong quantitative bound for the discretised ring theorem. In particular, we show that if $A \subset [1,2]$ is a $(\delta,\sigma)$-set, with $|A| = \delta^{-\sigma},$ then…
In this paper, we use Hilbert-Samuel multiplicity, Hilbert-Kunz multiplicity, and s-multiplicity to establish a sharp upper bound for the quotient of the generalized Milnor numbers and the Tjurina numbers for isolated hypersurface…
A detailed study of complex-space singularities of the two-dimensional incompressible Euler equation is performed in the short-time asymptotic r\'egime when such singularities are very far from the real domain; this allows an exact…
In this paper, we study Hausdorff and Fourier dimension from the point of view of effective descriptive set theory and Type-2 Theory of Effectivity. Working in the hyperspace $\mathbf{K}(X)$ of compact subsets of $X$, with $X=[0,1]^d$ or…
Consider an operator that takes the Fourier transform of a discrete measure supported in $\mathcal{X}\subset[-\frac 12,\frac 12)^d$ and restricts it to a compact $\Omega\subset\mathbb{R}^d$. We provide lower bounds for its smallest singular…
We consider four related problems. (1) Obtaining dimension estimates for the set of exceptional vantage points for the pinned Falconer distance problem. (2) Nonlinear projection theorems, in the spirit of Kaufman, Bourgain, and Shmerkin.…
We prove optimal concentration of measure for lifted functions on high dimensional expanders (HDX). Let $X$ be a $k$-dimensional HDX. We show for any $i\leq k$ and $f:X(i)\to [0,1]$: \[\Pr_{s\in X(k)}\left[\left|\underset{{t\subseteq…
A set $A$ of integers is said to be Schur if any two-colouring of $A$ results in monochromatic $x,y$ and $z$ with $x+y=z$. We study the following problem: how many random integers from $[n]$ need to be added to some $A\subseteq [n]$ to…
We determine the exact freezing threshold, r^f, for a family of models of random boolean constraint satisfaction problems, including NAE-SAT and hypergraph 2-colouring, when the constraint size is sufficiently large. If the…
We introduce and analyse infinite dimensional Wishart processes taking values in the cone $S^+_1(H)$ of positive self-adjoint trace class operators on a separable real Hilbert space $H$. Our main result gives necessary and sufficient…
In this paper we obtain new sufficient conditions for representation of a function as an absolutely convergent Fourier integral. Unlike those known earlier, these conditions are given in terms of belonging to weighted spaces. Adding weights…
The ability to resolve detail in the object that is being imaged, named by resolution, is the core parameter of an imaging system. Super-resolution is a class of techniques that can enhance the resolution of an imaging system and even…
We establish dimensional thresholds for dot product sets associated with compact subsets of translated paraboloids. Specifically, we prove that when the dimension of such a subset exceeds $ \frac{5}{4} = \frac{3}{2} - \frac{1}{4} $ in…
We study several distinct but related Fourier analytic variants of the well-known Kakeya and Furstenberg set problems in the plane. For example, given $0<s,t<1$, we call a set $K \subseteq \mathbb{R}^2$ an $(s,t)$-Kakeya set if there exists…
We investigate the sets of uniform limits $A(\bar{B}_n)$, $A(\bar{D}^I)$ of polynomials on the closed unit ball $\bar{B}_n$ of $\mathbb{C}^n$ and on the cartesian product $\bar{D}^I$ where $I$ is an arbitrary set and $\bar{D}$ is the closed…
We establish the following universality property in high dimensions: Let $X$ be a random vector with density in $\mathbb{R}^n$. The density function can be arbitrary. We show that there exists a fixed unit vector $\theta \in \mathbb{R}^n$…
In this paper the authors study set expansion in finite fields. Fourier analytic proofs are given for several results recently obtained by Solymosi, Vinh and Vu using spectral graph theory. In addition, several generalizations of these…