Moments of discrete measures with dense jumps induced by \(\beta\)-expansions
type: Item
instance of: scholarly article
title: Moments of discrete measures with dense jumps induced by \(\beta\)-expansions
zbMATH Open document ID: 1291.37008
published in: Journal of Mathematical Analysis and Applications
publication date: 2013-01-14T00:00:00Z
review text: Let \(\beta>1\) be a real number and \(\lceil \cdot \rceil\) be the ceiling function. The present paper studies the function \(\mu_{\beta}: [0,1] \rightarrow \mathbb{R}\) defined by \(\mu_{\beta}(\alpha):=\sum_{n=0}^\infty (\lceil \alpha(n+1)\rceil-\lceil \alpha(n)\rceil)/\beta^{n+1}\). In a previous work of the same author [J. Number Theory 133, No. 11, 3982--3994 (2013; \url{doi:10.1016/j.jnt.2012.11.008})], one sees that the function \(\mu_{\beta}\) is increasing, continuous at all irrational numbers and left-continuous but not right-continuous at all rational numbers. In this paper, the author shows that \(\mu_{\beta}\) is in fact a pure jump distribution, i.e., the Lebesgue-Stieltjes measure induced by \(\mu_{\beta}\) is discrete and the point masses are distributed over the whole rationals in \([0,1]\). More precisely, the author shows that the jump of \(\mu_{\beta}\) at each rational \(\alpha=p/q\in [0,1)\) with \(p,q\) being coprime, is \({\beta-1\over \beta(\beta^q-1)}\), and the Lebesgue-Stieltjes measure associated to \(\mu_{\beta}\) can be written as \(\sum_{0\leq p/q<1, \text{gcd}(p,q)=1}{\beta-1\over \beta(\beta^q-1)} \delta_{p/q}\), where \(\delta_t\) is the Dirac measure at \(t\) and the summation runs over all reduced rationals in \([0,1)\). Furthermore, the author proves that the \(m\)-th moment of this associated Lebesgue-Stieltjes measure is \(M_m={\beta-1 \over \beta} \sum_{r=0}^m {1 \over m+1-r} {m \choose r} B_rLi_{r-1}(\beta^{-1})\), where \(B_n\) is the \(n\)-th Bernoulli number and \(Li_n\) is the polylogarithm of order \(n\) defined by \(Li_n(z):=\sum_{k=1}^\infty z^k/k^n\). The asymptotics \(M_m= O(m^{1/4}e^{-2\sqrt{m\log \beta}})\) as \(m\to\infty\) is also proved.
reviewed by: Lingmin Liao
Mathematics Subject Classification ID: 37A45
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 37B10
Mathematics Subject Classification ID: 68R15
zbMATH DE Number: 6125361
zbMATH Keywords: moment
zbMATH Keywords: beta-expansion
zbMATH Keywords: Sturmian word
author: Do Yong Kwon
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.07.014
OpenAlex ID: W2044296615
cites work: On Some Singular Monotonic Functions Which Are Strictly Increasing
cites work: Q4000342
cites work: Riesz--Nágy singular functions revisited
cites work: The derivative of Minkowski's \(?(x)\) function
cites work: On the derivative of the Minkowski question mark function \(?(x)\)
cites work: A devil's staircase from rotations and irrationality measures for Liouville numbers
cites work: On the differentiability of certain saltus functions
cites work: The Cantor function
cites work: Addenda and corrigenda to “The Minkowski question mark function: explicit series for the dyadic period function and moments”
cites work: THE MOMENTS OF MINKOWSKI QUESTION MARK FUNCTION: THE DYADIC PERIOD FUNCTION
cites work: Semi-regular continued fractions and an exact formula for the moments of the Minkowski question mark function
cites work: A note on the moments of the Riesz-Nágy-Takács distribution
cites work: Jump Distributions on [ -1, 1 ] whose Orthogonal Polynomials have Leading Coefficients with Given Asymptotic Behavior
cites work: Sieved Orthogonal Polynomials and Discrete Measures with Jumps Dense in an Interval
cites work: Q4529547
cites work: On theβ-expansions of real numbers
cites work: Representations for real numbers and their ergodic properties
cites work: The orbit of a \(\beta\)-transformation cannot lie in a small interval
cites work: The natural extensions of β-transformations which generalize baker's transformations
cites work: Ordered orbits of the shift, square roots, and the devil's staircase
cites work: Q3518705
cites work: Q5558293
cites work: A problem seminar
cites work: Analytical depoissonization and its applications
cites work: Q3916824
DOI: 10.1016/J.JMAA.2012.07.014
Recommended article: DISCRETE MEASURES WITH DENSE JUMPS INDUCED BY STURMIAN DIRICHLET SERIES
Recommended article: Invariant densities for random \(\beta\)-expansions
Recommended article: A moment problem for random discrete measures
Recommended article: A moment problem for discrete nonpositive measures on a finite interval
Recommended article: Large deviations for discrete \(\beta\)-ensembles
Recommended article: Q3209358
Recommended article: Moments of the weighted Cantor measures
Recommended article: On Discrete Moments of Unbounded Order
Recommended article: Limit theorems for beta-Jacobi ensembles
Recommended article: Moments of the Gaussian β ensembles and the large-N expansion of the densities
description: scientific article; zbMATH DE number 6125361