On simultaneous digital expansions of polynomial values

Manfred Madritsch (1) , Thomas Stoll (1)

1 IECL - Institut Élie Cartan de Lorraine

Let $s_q$ denote the $q$-ary sum-of-digits function and let $P_1(X)$, $P_2(X) \in \mathbb{Z}[X]$ with $P_1(\mathbb{N}), P_2(\mathbb{N})\subset \mathbb{N}$ be polynomials of degree $h,l\geq 1$, $h\neq l$, respectively. In this note we show that $\left(s_q(P_1(n))/s_q(P_2(n))\right)_{n\geq 1}$ is dense in $\mathbb{R}^+$. This extends work by Stolarsky (1978) and Hare, Laishram and Stoll (2011).

Domaines: Théorie des nombres [math.NT]

Manfred Madritsch, Thomas Stoll. On simultaneous digital expansions of polynomial values. Acta Mathematica Hungarica, 2014, 143 (1), pp.192-200. ⟨10.1007/s10474-014-0394-4⟩. ⟨hal-00875103⟩