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

TL;DR

This paper proves that the joint distribution of base-$q$ digit sums and Ostrowski $eta$-representation digit sums in residue classes is uniformly distributed with a quantifiable error term, extending previous results to a broader class of $eta$.

Contribution

It establishes a uniform distribution result for the joint residue classes of two different digital sum functions, generalizing prior work to a wider range of $eta$-representations.

Findings

Proves uniform distribution with an error term for joint digital sums.

Extends previous results to Ostrowski representations with periodic continued fractions.

Provides bounds on the deviation from the expected distribution.

Abstract

Let $q$ be an integer $\geq 2$ and let $S_q(n)$ denote the sum of digits of $n$ in base $q$. For \[ \alpha=[0;\overline{1,m}],\ m\geq 2, \] let $S_{\alpha}(n)$ denote the sum of digits in the Ostrowski $\alpha$-representation of $n$. Let $m_1,m_2\geq 2$ 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_{\alpha}(n)\equiv a_2\pmod{m_2}\}| &=&\frac{N}{m_1m_2}+O(N^{1-\delta}). \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.