Related papers: Distribution of algebraic numbers

We prove a conjecture of Cuttler et al.~[2011] [A. Cuttler, C. Greene, and M. Skandera; \emph{Inequalities for symmetric means}. European J. Combinatorics, 32(2011), 745--761] on the monotonicity of \emph{normalized Schur functions} under…

We count algebraic numbers of fixed degree $d$ and fixed (absolute multiplicative Weil) height $\mathcal{H}$ with precisely $k$ conjugates that lie inside the open unit disk. We also count the number of values up to $\mathcal{H}$ that the…

We present quantitative versions of Bohr's theorem on general Dirichlet series $D=\sum a_{n} e^{-\lambda_{n}s}$ assuming different assumptions on the frequency $\lambda:=(\lambda_{n})$, including the conditions introduced by Bohr and…

We introduce and study a generalization $s_{(\mu|\lambda)}$ of the Schur functions called the almost symmetric Schur functions. These functions simultaneously generalize the finite variable key polynomials and the infinite variable Schur…

We investigate a variant of Wirsing's problem on approximation to a real number by real algebraic numbers of degree exactly $n$. This has been studied by Bugeaud and Teulie. We improve their bounds for degrees up to $n=7$. Moreover, we…

Mahler's problem asks for the largest possible value of the Mahler measure, normalized by the $L_2$ norm, of a polynomial with $\pm1$ coefficients and large degree. We establish a new record value in this problem exceeding $0.95$ by…

Measures with values in the set of sesquilinear forms on a subspace of a Hilbert space are of interest in quantum mechanics, since they can be interpreted as observables with only a restricted set of possible measurement preparations. In…

We study the asymptotic distribution of critical values of random holomorphic `polynomials' s_n on a Kaehler manifold M as the degree n tends to infinity. By `polynomial' of degree n we mean a holomorphic section of the nth power of a…

Given a sequence of polynomials $Q_n$ of degree $n$ with zeros on $[-1,1]$, we consider the triangular table of derivatives $Q_{n, k}(x)=d^k Q_n(x) /d x^k$. Under the assumption that the sequence $\{Q_n\}$ has a weak* limiting zero…

In 1908 Thue (1) showed that algebraic numbers of the special form $\xi =\sqrt[n]{\frac{a}{b}}$ can, for every positive $\epsilon$, only be sharply approximated by finitely many rational numbers $\frac{p}{q}$ with the following inequality…

We give a bound on the dimension of the Schur multiplier of a finite dimensional nilpotent Lie algebra which sharpens the earlier known bounds.

Let $A$ be a finite dimensional algebra over an algebraically closed field $k$. Assume $A$ is a basic connected and triangular algebra with $n$ pairwise non-isomorphic simple modules. We consider the {\em Coxeter transformation} $\phi_A(T)$…

Let $n_1 < n_2 < \cdots < n_N$ be non-negative integers. In a private communication Brian Conrey asked how fast the number of real zeros of the trigonometric polynomials $T_N(\theta) = \sum_{j=1}^N {\cos (n_j\theta)}$ tends to $\infty$ as a…

This article addresses an equidistribution problem concerning the zeros of systems of random holomorphic sections of positive line bundles on compact K\"{a}hler manifolds and random polynomials on $\mathbb{C}^{m}$ in the setting of the…

The coefficients of the regular continued fraction for random numbers are distributed by the Gauss-Kuzmin distribution according to Khinchin's law. Their geometric mean converges to Khinchin's constant and their rational approximation speed…

We establish a rather unexpected and simple criterion for the boundedness of Schur multipliers $S_M$ on Schatten $p$-classes which solves a conjecture proposed by Mikael de la Salle. Given $1 < p < \infty$, a simple form our main result…

We propose an extension of the theory of parity sheaves, which allows for non-locally constant sheaves along strata. Our definition is tailored for proving the existence of (proper, quasihereditary, etc) stratifications of…

In order to find a way to have a better formulation for numerical evolution of the Einstein equations, we study the propagation equations of the constraints based on the Arnowitt-Deser-Misner formulation. By adjusting constraint terms in…

We establish the limiting distribution of $\frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{n\le x}\alpha(n)$ where $\alpha$ is a Steinhaus random multiplicative function, answering a question of Harper. The distributional convergence is proved…

We study the effective equidistribution of certain infinite homogeneous measures and related counting problems through mixing. In this way, we obtain smooth versions of counting theorems studied by Oh-Shah and later by Kelmer-Kontorovich…