Joint distribution in residue classes of the base-$q$ and Ostrowski digital sums

Abstract

Let $q$ be an integer $\ge 2$ and let $S_q(n)$ denote the sum of digits of $n$ in base $q$. For \[ α=[0;\overline{1,m}],\ m\geq 2, \] let $S_{\alpha }(n)$ denote the sum of digits in the Ostrowski $\alpha$-representation of $n$. Let $m_1,m_2\ge 2$ be integers with $$\gcd (q-1,m_1)=\gcd (m,m_2)=1.$$ We prove that there exists $\delta >0$ such that for all integers $a_1,a_2$, \begin{eqnarray*} &&|\{0\leq n<N: S_{q}(n)\equiv a_1\pmod{m_1},\ S_α(n)\equiv a_2\pmod{m_2}\}| &=&\frac{N}{m_1m_2}+O(N^{1-δ}). \end{eqnarray*} The asymptotic relation implied by this equality was proved by Coquet, Rhin & Toffin and the equality was proved for the case $\alpha =[\overline{1}]$ by Spiegelhofer.

18 pages

References in corpus (1)

• Pseudorandomness of the Ostrowski sum-of-digits function