Related papers: A quantitative framework for sets of exact approxi…

Let $(\Sigma, \sigma)$ be the one-sided shift space with $m$ symbols and $R_n(x)$ be the first return time of $x\in\Sigma$ to the $n$-th cylinder containing $x$. Denote $$E^\varphi_{\alpha,\beta}=\left\{x\in\Sigma:…

We define $\Psi$-autoreducible sets given an autoreduction procedure $\Psi$. Then, we show that for any $\Psi$, a measurable class of $\Psi$-autoreducible sets has measure zero. Using this, we show that classes of cototal, uniformly…

The \emph{Diophantine exponent} of an action of a group on a homogeneous space, as defined by Ghosh, Gorodnik, and Nevo, quantifies the complexity of approximating the points of the homogeneous space by the points on an orbit of the group.…

The seminal work of Kurzweil (1955) provides for any fixed badly approximable $\alpha$ and monotonically decreasing $\psi$ a Khintchine-type statement on the set of the inhomogeneous real parameters $\gamma$ for which $\lVert n \alpha +…

In this article we will discuss a few aspects of the spacetime description of matter and fields. In Section:1 we will discuss the completeness of real numbers in the context of an alternate definition of the straight line as a geometric…

We show that whenever $\delta>0$, $\eta$ is real and constants $\lambda_i$ satisfy some necessary conditions, there are infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality $|\lambda_1p_1 + \lambda_2p_2 +…

Theorems of Khintchine, Groshev, Jarn\'ik, and Besicovitch in Diophantine approximation are fundamental results on the metric properties of $\Psi$-well approximable sets. These foundational results have since been generalised to the…

The quantum relative entropy is a fundamental quantity in quantum information science, characterizing the distinguishability between two quantum states. However, this quantity is not additive in general for correlated quantum states,…

We prove new quantitative Schmidt-type theorem for Diophantine approximations with restraint denominators on fractals (more precisely, on $M_0$-sets). Our theorems introduce a sharp balance condition between the growth rate of the sequence…

We present a survey of ergodic theorems for actions of algebraic and arithmetic groups recently established by the authors, as well as some of their applications. Our approach is based on spectral methods employing the unitary…

For $p \in (1,N)$ and $\Omega \subseteq \mathbb{R}^N$ open, the Beppo-Levi space $\mathcal{D}^{1,p}_0(\Omega)$ is the completion of $C_c^{\infty}(\Omega)$ with respect to the norm $\left( \int_{\Omega}|\nabla u|^p \right)^ \frac{1}{p}.$…

We present a new argument in the study of positive entropy measures for higher rank diagonalisable actions. The argument relies on a quantitative form of recurrence along unipotent directions (that are not known to preserve the measure).…

For regular continued fraction, if a real number $x$ and its rational approximation $p/q$ satisfying $|x-p/q|<1/q^2$, then, after deleting the last integer of the partial quotients of $p/q$, the sequence of the remaining partial quotients…

We introduce a new information-theoretic formulation of quantum measurement uncertainty relations, based on the notion of relative entropy between measurement probabilities. In the case of a finite-dimensional system and for any approximate…

For a fixed integer $n$, we study the question whether at least one of the numbers $\Re X\omega^k$, $1\leq k\leq n$, is $\varepsilon$-close to an integer, for any possible value of $X\in\mathbb{C}$, where $\omega$ is a primitive $n$th root…

We present a comprehensive framework for the study of the size and large intersection properties of sets of limsup type that arise naturally in Diophantine approximation and multifractal analysis. This setting encompasses the classical…

We study different estimators of the radius of convergence of the Taylor series of the pressure in finite density QCD. We adopt the approach in which the radius of convergence is estimated first in a finite volume, and the infinite-volume…

In this paper we develop a general framework of badly approximable points in a metric space $X$ equipped with a $\sigma$-finite doubling Borel regular measure $\mu$. We establish that under mild assumptions the $\mu$-measure of the set of…

The Mittag-Leffler function $E_{\alpha}$ being a natural generalization of the exponential function, an infinite-dimensional version of the fractional Poisson measure would have a characteristic functional \[ C_{\alpha}(\phi)…

This paper is an erratum to our paper, entitled "On an application of Guth-Katz theorem", Math. Res. Lett. 18 (2011), no. 4, 691-697. Let $F$ be the real or complex field and $\omega$ a non-degenerate skew-symmetric bilinear form in the…