On the exponential sum with the sum of digits of hereditary base \(b\) notation (Q2926261)

Let \(b\geq 2\) be an integer and \(w_b(n)\) be the sum of digits of the non-negative integer \(n\) written in hereditary base \(b\) notation. To obtain the hereditary base \(b\) notation of a non-negative integer \(n\) we write \(n\) in base \(b\), and then we write all the exponents in base \(b\), etc. until there appear only the numbers \(0,1,\dots,b\). In the paper under review, the author studies the exponential sum NEWLINE\[NEWLINE S_b(t,N):=\sum_{n=0}^{N-1}\mathrm{e}^{2\pi iw_b(n)t}, NEWLINE\]NEWLINE where \(N\) is a positive integer and \(t\) is a non-integer real. The author gives optimal upper bounds for \(S_b(t,N)\), distinguishing between rational and irrational \(t\). When \(t\) is irrational, the author assumes \(b\geq 3\) and proves validity of \(S_b(t,N)=O_{b,t,\varepsilon}(N^{C_{b,t}+\varepsilon})\) for \(N\geq 1\), and for each \(\varepsilon>0\), where \(C_{b,t}\) is the best possible constant given by NEWLINE\[NEWLINE C_{b,t}=\max\left\{0,\log_b\Big|\frac{\sin(\pi(b-1)t)}{\sin(\pi t)}\Big|\right\}\in[0,1[. NEWLINE\]NEWLINE For some technical problems explained by author, the argument presented in the paper fails to get desired bounds for the case \(b=2\) and \(t\) is irrational, and the author leaves this gap as an open problem. Anyway, for each \(b\geq 2\) and each non-integer real \(t\) the author deduces the approximation \(S_b(t,N)=o(N)\) as \(N\to\infty\), from which by means of Weyl criterion, it implies that for each positive integer \(m\), the sequence \((w_b(n))_{n\geq 0}\) is uniformly distributed modulo \(m\), and also for each irrational real \(\alpha\), the sequence \((\alpha w_b(n))_{n\geq 0}\) is uniformly distributed modulo \(1\).