On simultaneous digital expansions of polynomial values (Q397020)

Let \(q\geq 2\) and \(s_q\) denote the \(q\)-ary sum-of-digits function. Let \(P_1\) and \(P_2\in{\mathbb Z}[X]\) be polynomials of distinct positive degrees with \(P_1(\mathbb N)\subset\mathbb N\) and \(P_2(\mathbb N)\subset\mathbb N\). It is shown that \((s_q(P_1(n))/s_q(P_2(n)))_{n\geq 1}\) is dense in \({\mathbb R}^+\). This extends work by \textit{K. B. Stolarsky} [Proc. Am. Math. Soc. 71, 1--5 (1978; Zbl 0391.10011)] and \textit{K. G. Hare}, \textit{S. Laishram} and \textit{T. Stoll} [Proc. Am. Math. Soc. 139, No. 1, 39--49 (2011; Zbl 1233.11007)].