Treffer: DISTRIBUTION OF DIGITS IN INTEGERS: BESICOVITCH–EGGLESTON SUBSETS OF ${\mathbb N}$: Distribution of digits in integers: Besicovitch-Eggleston subsets of \(\mathbb N\)

For a positive integer \(n\), denote the \(N\)-ary expansion of \(n\) by \(n=d_0(n)+d_1(n)N +\dots+d_{\gamma(n)}(n)N^{\gamma(n)}\), where \(d_i(n)\in\{0,1,\dots,N-1\}\) and \(d_{\gamma(n)}(n)\neq 0\). For a probability vector \(p=(p_0,\dots,p_{N-1})\), the \(r\) approximative discrete Besicovitch-Eggleston set \(B_r(p)\) is defined by \[ B_r(p)=\left\{n\in\mathbb{N}| \left| \frac{| \{0\leq k\leq\gamma(n)\mid d_k(n)=i\}| }{\gamma(n)+1}-p_i\right| \leq r, \forall i\right\}. \] For a set \(E\subseteq\mathbb{N}\), let \(N_n(E)=| \{1,\dots,n\}\cap E| \). The upper and lower fractal dimensions of \(E\) are defined by \(\underline{\dim}(E)=\liminf_n\frac{\log N_n(E)}{\log n}\) and \(\overline{\dim}(E)=\limsup_n\frac{\log N_n(E)}{\log n}\), respectively. The exponent of convergence of \(E\) is defined by \(\delta(E)=\sum\left\{t\geq 0\colon \sum_{n\in E}\frac{1}{n^t}=\infty\right\}\). The purpose of this paper is to show that these three natural fractional dimensions of \(E\) satisfy a formula similar to the Besicovitch-Eggleston Theorem, proving that \[ \lim_{r\searrow 0}\underline{\dim}(B_r(p))=\lim_{r\searrow 0}\overline{\dim}(B_r(p))=\lim_{r\searrow 0}\delta(B_r(p)) = -\frac{\sum_i p_i\log p_i}{\log N}. \] An application to the theory of normal numbers is to strengthen a result by \textit{B. D. Craven} [J. Aust. Math. Soc. 5, 325--330 (1965; Zbl 0144.27504)].