Related papers: Fourier uniformity on subspaces

We show that if a closed discrete subset $A \subseteq \mathbf{R}^d$ is denser than a certain critical threshold, then $A$ is a Fourier uniqueness set, while if $A$ is sparser, then uniqueness fails and one can prescribe arbitrary values for…

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 the following conjecture (from Huang, et al): Let $\Delta^+$ denote the upper half disc in $\mathbb{C}$ and let $\gamma = ( - 1, 1)$ (viewed as an interval in the real axis in $\mathbb{C}$). Assume that $F$ is a holomorphic…

Let V be a vector space of dimension n over the finite field F_q, where q is odd, and let Symm(V) denote the space of symmetric bilinear forms defined on V x V. We investigate constant rank r subspaces of Symm(V) in this paper. We have…

Dvir and Moran proved the following upper bound for the size of a family $\mbox{$\cal F$}$ of subsets of $[n]$ with $\mbox{Vdim}(\mbox{$\cal F$} \Delta \mbox{$\cal F$})\leq d$. Let $d\leq n$ be integers. Let $\mbox{$\cal F$}$ be a family of…

These are notes of a talk based on the work arXiv:1212.3630 joint with A. Aizenbud. Let V be a finite-dimensional vector space over a local field F of characteristic 0. Let f be a function on V of the form $f(x)= \psi (P(x))$, where P is a…

There exists an absolute constant $\delta > 0$ such that for all $q$ and all subsets $A \subseteq \mathbb{F}_q$ of the finite field with $q$ elements, if $|A| > q^{2/3 - \delta}$, then \[ |(A-A)(A-A)| = |\{ (a -b) (c-d) : a,b,c,d \in A\}| >…

We study the finite dimensional spaces $V$ which are invariant under the action of the finite differences operator $\Delta_h^m$. Concretely, we prove that if $V$ is such an space, there exists a finite dimensional translation invariant…

Let $A$ be a simple algebra over a field $F$. Under a mild cardinality assumption on $F$, we determine the greatest possible dimension for an $F$-affine subspace of $A$ that is included in the group of units $A^\times$, and we describe the…

We define a natural notion of higher order stability and show that subsets of $\mathbb{F}_p^n$ that are tame in this sense can be approximately described by a union of low-complexity quadratic varieties, up to linear error. This generalizes…

Let $\mathbb{D}$ be a division ring and $\mathbb{F}$ be a subfield of the center of $\mathbb{D}$ over which $\mathbb{D}$ has finite dimension $d$. Let $n,p,r$ be positive integers and $\mathcal{V}$ be an affine subspace of the…

The arithmetic regularity lemma for $\mathbb{F}_p^n$, proved by Green in 2005, states that given a subset $A\subseteq \mathbb{F}_p^n$, there exists a subspace $H\leq \mathbb{F}_p^n$ of bounded codimension such that $A$ is Fourier-uniform…

Suppose that A is a subset of F_2^n of density as close to 1/3 as possible. We show that the A(F_2^n)-norm (that is the sum of the absolute values of the Fourier transform) of the characterstic function of A is bounded below by an absolute…

We study the fine-grained uniform convergence behavior of halfspaces beyond worst-case VC bounds. For inhomogeneous halfspaces in $\mathbb{R}^d$ with $d\ge 2$, we show that standard first-order VC bounds are essentially tight: even…

The entanglement entropy of an arbitrary spacetime region $A$ in a three-dimensional conformal field theory (CFT) contains a constant universal coefficient, $F(A)$. For general theories, the value of $F(A)$ is minimized when $A$ is a round…

Let $\mathbb{F}$ be a field, and $n \geq r>0$ be integers, with $r$ even. Denote by $\mathrm{A}_n(\mathbb{F})$ the space of all $n$-by-$n$ alternating matrices with entries in $\mathbb{F}$. We consider the problem of determining the…

We give a sufficient condition for the Fourier dimension of a countable union of sets to equal the supremum of the Fourier dimensions of the sets in the union, and show by example that the Fourier dimension is not countably stable in…

Let $f$ be a Hecke-Maass cusp form for $\rm SL_2(\mathbb{Z})$ with Laplace eigenvalue $\lambda_f(\Delta)=1/4+\mu^2$ and let $\lambda_f(n)$ be its $n$-th normalized Fourier coefficient. It is proved that, uniformly in $\alpha, \beta \in…

We systematically find conditions which yield locally uniform convergence in the Fourier inversion formula in one and higher dimensions. We apply the gained knowledge to the complex inversion formula of the Laplace transform to extend known…

We consider the function $x^{-1}$ that inverses a finite field element $x \in \mathbb{F}_{p^n}$ ($p$ is prime, $0^{-1} = 0$) and affine $\mathbb{F}_{p}$-subspaces of $\mathbb{F}_{p^n}$ such that their images are affine subspaces as well. It…