Moments of discrete measures with dense jumps induced by \(\beta\)-expansions
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.
scientific article; zbMATH DE number 6125361
scientific article