Related papers: On Simon's Hausdorff Dimension Conjecture

A graph is said to be diameter-$k$-critical if its diameter is $k$ and removal of any of its edges increases its diameter. A beautiful conjecture by Murty and Simon, says that every diameter-2-critical graph of order $n$ has at most…

In this paper we show that the Hausdorff dimension of the set of singular pairs is 4/3. We also show that the action of diag(e^t,e^t,e^{-2t}) on SL(3,R)/SL(3,Z) admits divergent trajectories that exit to infinity at arbitrarily slow…

In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the Zeilberger--Bressoud $q$-Dyson constant term identity. The non-zero part of Kadell's conjecture is a constant term identity indexed by a weak…

In the late seventies, Sullivan showed that for a convex cocompact subgroup $\Gamma$ of $\operatorname{SO}^\circ(n,1)$ with critical exponent $\delta>0$, any $\Gamma$-conformal measure on $\partial \mathbb{H}^n$ of dimension $\delta$ is…

Suppose that $\Sigma$ is a hyperbolic surface and $f:\mathbb R_+\to\mathbb R_+$ a monotonic function. We study the closure in the projective tangent bundle $PT\Sigma$ of the set of all geodesics $\gamma$ satisfying $I(\gamma,\gamma)\leq…

The $n$-th Christoffel function for a point $z_0\in\mathbb C$ and a finite measure $\mu$ supported on a Jordan arc $\Gamma$ is \[ \lambda_n(\mu,z_0)=\inf\left\{\int_\Gamma |P|^2d\mu\mid P\text{ is a polynomial of degree at most }n\text{ and…

It is well known from the Perron-Frobenius theory that the spectral gap of a positive square matrix is positive. In this paper, we give a more quantitative characterization of the spectral gap. More specifically, using a complex extension…

We study the combination of two o-minimal extensions of the theory of real closed fields: one by a T-convex subring and the other by a T-derivation. Let T be a complete, model complete o-minimal extension of RCF. We show that the combined…

In 1972, A. J. Hoffman proved his celebrated theorem concerning the limit points of spectral radii of non-negative symmetric integral matrices less than $\sqrt{2+\sqrt{5}}$. In this paper, after giving a new version of Hoffman's theorem, we…

Let $\psi:\mathbb{N}\rightarrow\mathbb{R}_+$ be a monotonically non-increasing function, and let $\psi_v:\mathbb{N}\rightarrow\mathbb{R}_+$ be defined by $\psi_v(q)=1/q^v$. In this article, we consider self-similar sets whose iterated…

In their seminal paper Erd\H{o}s and Szemer\'edi formulated conjectures on the size of sumset and product set of integers. The strongest form of their conjecture is about sums and products along the edges of a graph. In this paper we show…

We prove a sharp, quantitative analogue of Helgason's conjecture at the level of distributions: For a semisimple Lie group $G$ of real rank one, Poisson transforms map a Sobolev space on $P\backslash G$ boundedly with closed range to an…

Let $X=\{ X_n\}_{n\in \mathbb{Z}}$ be zero-mean stationary Gaussian sequence of random variables with covariance function $\rho$ satisfying $\rho(0)=1$. Let $\varphi:\mathbb{R}\to\mathbb{R}$ be a function such that…

Confining a quantum particle in a compact subinterval of the real line with Dirichlet boundary conditions, we identify the connection of the one-dimensional fractional Schr\"odinger operator with the truncated Toeplitz matrices. We…

In this paper, we prove a Szeg\"{o} type limit theorem on $\ell^2(\ZZ^d)$. We consider operators of the form $H=\Delta+V$, $V$ multiplication by a positive sequence $\{V(n), n \in \ZZ^d\}$ with $V(n) \rightarrow \infty, |n| \rightarrow…

In this note we will describe a simple and practical approach to get rigorous bounds on the Hausdorff dimension of limits sets for some one dimensional Markov iterated function schemes. The general problem has attracted considerable…

Let $\mathcal{H}=-\Delta_{\mathbb{H}}+V$ be the Schr\"odinger operator on the Heisenberg group $\mathbb{H}^n$, where $\Delta_{\mathbb{H}}$ is the full laplacian on $\mathbb{H}^n$ and $V$ is a positive smooth potential, bounded below and…

The Bernoulli convolution with parameter $\lambda\in(0,1)$ is the probability measure $\mu_\lambda$ that is the law of the random variable $\sum_{n\ge0}\pm\lambda^n$, where the signs are independent unbiased coin tosses. We prove that each…

Let $u$ be a harmonic function in the unit ball $B_1 \subset \mathbb R^n$, normalized so that its gradient has magnitude at most 1 on the unit ball. We show that if the gradient of $u$ is $\epsilon$-small in size on a set $E\subset B_{1/2}$…

Additive combinatorics asks for lower bounds on sumsets and restricted sumsets over finite fields. Central examples are the Cauchy-Davenport theorem and the Erd\H{o}s-Heilbronn conjecture. In this note, we develop Das's linear algebraic…