Decimals and almost-periodicity (Q854697)

An arithmetical function \(f\) is called almost-periodic, if for any \(\varepsilon > 0\) there exists a \(\mathbb C\)-linear combination of exponentials \(e^{2\pi i \alpha n}\), \(\alpha \in \mathbb R\), \(\varepsilon\)-near to \(f\) with respect to the semin-norm \(\|f\| = \lim\sup_{N\to\infty} \frac1N \cdot \sum_{n\leq x} |f(n)|\). There are many results on almost-periodic functions which are multiplicative or additive. The author is motivated by the search for other almost-periodic arithmetical functions. He proves: For fixed \(g\geq 2\), \(g\in \mathbb N\), let \(e \in \{1,2,\dots,g-1\}\) be a fixed digit. Let \((a_n)_{n\geq 1}\) be a sequence of real numbers, satisfying \[ a_n \sim \gamma\alpha^n, \text{ where } \gamma,\alpha > 0, \text{ and } \log \alpha \text{ is irrational}, \] then the function \(f\), defined by \[ f(n)=\begin{cases} 1, & \text{ if } a_n\not=0 \text{ and the leading digit is }e, \\ 0 & \text{ otherwise},\end{cases} \] is almost-periodic with mean-value \(\log(1+\frac1e)\). Under suitable assumptions this result implies a corresponding result, where the sequence \(a_n\) is solution of a linear recurrence. An important tool for the proof of the first result is the uniform distribution of the sequence \((\log(\gamma\alpha^n))_n\).