Related papers: On syndetic Riesz sequences

Let $\lambda$ be a positive number, and let $(x_j:j\in\mathbb Z)\subset\mathbb R$ be a fixed Riesz-basis sequence, namely, $(x_j)$ is strictly increasing, and the set of functions $\{\mathbb R\ni t\mapsto e^{ix_jt}:j\in\mathbb Z\}$ is a…

We randomly construct various subsets $\Lambda$ of the integers which have both smallness and largeness properties. They are small since they are very close, in various meanings, to Sidon sets: the continuous functions with spectrum in…

Let K be an algebraically closed field of prime characteristic p, let N be a positive integer, let f be a self-map on the algebraic torus T=G_m^N defined over K, let V be a curve in T defined over K, and let x be a K-point of T. We show…

Starting with topological field theories we investigate the Ray-Singer analytic torsion in three dimensions. For the lens Spaces L(p;q) an explicit analytic continuation of the appropriate zeta functions is contructed and implemented. Among…

We prove an asymptotically tight lower bound on $|A+\lambda A|$ for $A\subset \mathbb{C}$ and algebraic integer $\lambda$. The proof combines strong version of Freiman's theorem, structural theorem on dense subsets of a hypercubic lattice…

We study the observability of the Schr\"odinger equation on the $d$-dimensional torus $\mathbb T^d$, $d \geq 1$, from an open subset $\omega \subset \mathbb T^d$. Our first main result establishes a quantitative observability estimate for…

We prove that for any convex polytope $\Omega \subset \mathbb{R}^d$ which is centrally symmetric and whose faces of all dimensions are also centrally symmetric, there exists a Riesz basis of exponential functions in the space $L^2(\Omega)$.…

We prove a Krieger like embedding theorem for asymptotically expansive systems with the small boundary property. We show that such a system $(X; T)$ embeds in the $K$-full shift with $h_{top}(T) < \log K $ and $\sharp Per_n(X; T) \leq…

For every set $S$ of finite measure in $\mathbb{R}$ we construct a discrete set of real frequencies $\Lambda$ such that the exponential system $\{\exp(i\lambda t),\lambda\in\Lambda\}$ is a frame in $L^2(S)$

Renling Jin proved that if A and B are two subsets of the natural numbers with positive Banach density, then A+B is piecewise syndetic. In this paper, we prove that, under various assumptions on positive lower or upper densities of A and B,…

We show an extension of a probabilistic result of Marcus, Spielman, and Srivastava, which resolved the Kadison-Singer problem, for block diagonal positive semidefinite random matrices. We use this result to show several selector results,…

In 1971 Cusick proved that every real number $x\in[0,1]$ can be expressed as a sum of two continued fractions with no partial quotients equal to $1$. In other words, if we define a set $$ S(k):= \{ x\in[0,1] : a_n(x) \geq k \text{ for all }…

For a large class of tilings, including the Penrose tiling in two dimension as well as the icosahedral ones in 3 dimension, the continuous hull of such a tiling inherits a minimal lamination structure with flat leaves and a transversal…

Can we find a self-similar set on the line with positive Lebesgue measure and empty interior? Currently, we do not have the answer for this question for deterministic self-similar sets. In this paper we answer this question negatively for…

Here, we study some measures that can be represented by infinite Riesz products of 1-periodic functions and are related to the doubling map. We show that these measures are purely singular continuous with respect to Lebesgue measure and…

Given a frequency $\lambda=(\lambda_n)$, we study when almost all vertical limits of a $\mathcal{H}_1$-Dirichlet series $\sum a_n e^{-\lambda_ns}$ are Riesz-summable almost everywhere on the imaginary axis. Equivalently, this means to…

We prove that Tietze Extension does not always exist in constructive mathematics if closed sets on which the function we are extending are defined as sequentially closed sets. Firstly, we take a discrete metric space as our topological…

In this paper we show that every set $A \subset \mathbb{N}$ with positive density contains $B+C$ for some pair $B,C$ of infinite subsets of $\mathbb{N}$, settling a conjecture of Erd\H{o}s. The proof features two different decompositions of…

Let $\M$ be a smooth connected non-compact manifold endowed with a smooth measure $\mu$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $\mu$. We show that if $L$ satisfies,…

Let $\mu$ be a measure in $\mathbb R^d$ with compact support and continuous density, and let $$ R^s\mu(x)=\int\frac{y-x}{|y-x|^{s+1}}\,d\mu(y),\ \ x,y\in\mathbb R^d,\ \ 0<s<d. $$ We consider the following conjecture: $$ \sup_{x\in\mathbb…