Related papers: On Mahler's transcendence measure for $e$
For a finitely generated algebra over a field, the transcendence degree is known to be equal to the Krull dimension. The aim of this paper is to generalize this result to algebras over rings. A new definition of the transcendence degree of…
We give a revised version of Schmidt's treatment of forms in many variables, which allows us to prove a Hasse principle under more lenient conditions on the number of variables than what had previously been thought possible with these…
Magnitude is a numerical invariant of finite metric spaces, recently introduced by T. Leinster, which is analogous in precise senses to the cardinality of finite sets or the Euler characteristic of topological spaces. It has been extended…
Let $M(\alpha)$ denote the (logarithmic) Mahler measure of the algebraic number $\alpha$. Dubickas and Smyth, and later Fili and the author, examined metric versions of $M$. The author generalized these constructions in order to associate,…
It has been known for over 70 years that there is an asymptotic transition of Charlier polynomials to Hermite polynomials. This transition, which is still presented in its classical form in modern reference works, is valid if and only if a…
The aim of this paper is to give a new proof of the complete characterization of measures for which there exist a solution of the Dirichlet problem for the complex Monge-Ampere operator in the set of plurisubharmonic functions with finite…
K. Mahler introduced the concept of perfect systems in the theory of simultaneous Hermite-Pad\'{e} approximation of analytic functions. Recently, we proved that Nikishin systems, generated by measures with bounded support and…
The leading asymptotic behaviour as $t\to \infty$ of the celebrated Riemann zeta function $\zeta(s), \ s = \sigma + it, \quad 0<\sigma<1, \quad t>0 , \ t\to\infty,$ can be expressed in terms of a transcendental sum. The sharp estimation of…
In this work we apply the techniques that were developed in [Lalin: An algebraic integration for Mahler measure] in order to study several examples of multivariable polynomials whose Mahler measure is expressed in terms of special values of…
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…
Infinite determinantal measures introduced in this note are inductive limits of determinantal measures on an exhausting family of subsets of the phase space. Alternatively, an infinite determinantal measure can be described as a product of…
In this article we obtain new irrationality measures for values of functions which belong to a certain class of hypergeometric functions including shifted logarithmic functions, binomial functions and shifted exponential functions. We…
In the present paper we prove that for any open connected set $\Omega\subset\mathbb{R}^{n+1}$, $n\geq 1$, and any $E\subset \partial \Omega$ with $\mathcal{H}^n(E)<\infty$, absolute continuity of the harmonic measure $\omega$ with respect…
We prove new cases of reasonable bounds for the polynomial Szemer\'{e}di theorem both over $\mathbb{Z}/N\mathbb{Z}$ with $N$ prime and over the integers. In particular, we prove reasonable bounds for Szemer\'edi's theorem in the integers…
In a recent work [JNT \textbf{129}, 2154 (2009)], Gun and co-workers have claimed that the number $\,\log{\Gamma(x)} + \log{\Gamma(1-x)}\,$, $x$ being a rational number between $0$ and $1$, is transcendental with at most \emph{one} possible…
In this paper, we review the basic properties of measures vanishing at infinity and prove a version of the Riemann--Lebesgue lemma for Fourier transformable measures.
Here we give a lower bound of the Mahler measure on a set of polynomials that are "almost" reciprocal. Here "almost" reciprocal means that the outermost coefficients of each polynomial mirror each other in proportion, while this pattern…
We review some results of calculations, having the property of maximal transcendentality.
Given a frequency $\lambda$, we study general Dirichlet series $\sum a_n e^{-\lambda_n s}$. First, we give a new condition on $\lambda$ which ensures that a somewhere convergent Dirichlet series defining a bounded holomorphic function in…
Coherent superpositions are one of the hallmarks of quantum mechanics and are vital for any quantum mechanical device to outperform the classically achievable. Generically, superpositions are verified in interference experiments, but…