On the arithmetic structure of sets characterized by sum of digits properties (Q678391)

The authors are interested in the number-theoretic structure of the sets \[ U_\Gamma(N)= \{n\leq N;\;S(n)\equiv\Gamma\bmod m\} \] where \(S(n)\) denotes the sum of digits of the \(g\)-ary expansion of \(n\), and \(\text{gcd} (m,g-1)=1\). They prove four theorems, giving in particular: the size of sets \[ \{(a,b);\;S(a+b)\equiv\Gamma\bmod m\}, \] where \((a,b)\in{\mathcal A}\times{\mathcal B}\subset\{1,2,\dots, N\}^2\); an Erdös-Kac like theorem for the elements of \(U_\Gamma(N)\); and lower bounds for \(\max_{n\in U_\Gamma(N)}\omega(n)\) and \(\max_{n\in U_\Gamma(N)} \Omega(g,n)\), where \(\Omega(g,n)= \sum_{\substack{ p^\alpha\parallel n\\p\nmid g}} \alpha\). As a consequence of the first theorem they can show that the numbers \(S(p+q)\), where \(p,q\leq N\) are prime numbers, are well-distributed modulo \(m\). The crucial tools are a formula of \textit{A. O. Gelfond} [Acta Arith. 13, 259-265 (1968; Zbl 0155.09003)] and a nice number-theoretic-combinatorial lemma due to \textit{P. Erdös}, \textit{C. Pomerance}, \textit{A. Sárközy} and \textit{C. L. Stewart} [J. Number Theory 44, 93-104 (1993; Zbl 0780.11040)].