Sum-of-digits function for certain nonstationary bases
Let \(\mathcal Q=(q_n)_{n=1}^\infty\) be a strictly increasing sequence of positive integers with \(q_1=1\). Each natural \(n\) can be represented in the form \(n=\sum_{j=1}^k\epsilon_j q_j\), where \(q_k\leq n\leq q_{k+1}\) and \(\epsilon_k=[n/q_k]\), \(\epsilon_j=[(n-\sum_{i=j+1}^k\epsilon_i q_i)/q_j]\) (\(1\leq j<k\)). For every positive integer \(N\) put \(S(N)=\sum_{n<N}\sum_{j=1}^k\epsilon_j\), which is called the sum-of-digits function for the base \(\mathcal Q\). In this paper the author provides asymptotic formulas for \(S(N)\) for the case of Cantor and Ostrowski expansions. When \(q_1=1\) and \(q_n=a_1\dots a_{n-1}\) (\(n\geq 2\)), the expansion \(n=\sum_{j=1}^k\epsilon_j q_j\) is called Cantor expansion. If the condition \(\max(a_1,a_2,\dots,a_n)=o(\sum_{k=1}^n a_k)\) (\(n\to\infty\)) is satisfied, then \[ S(N)=({1\over 2}\sum_{k=1}^{n-1}(a_k-1))\dot N+O(\max(a_1,a_2,\dots,a_n)\dot N). \] When \(q_n\) is the denominator of the \(n\)th convergent of the continued fraction expansion of an irrational \(\alpha=[a_1,a_2,\dots]\), the expansion above is called Ostrowski expansion. If the same condition as above is satisfied, then \[ S(N)=(\sum_{k=1}^{n-1}R_k\alpha_k)\dot N+O(\max_{1\leq k\leq n}a_k\dot N), \] where \(R_k=(a_k)(q_{k+1}-q_k+q_{k-1})\) and \(\alpha_k=|q_k-p_k|\) (\(k=1,2,\dots\)).
scientific article; zbMATH DE number 1408564