Joint distribution of the base-q and Ostrowski digital sums

Divyum Sharma, Department of Pure Mathematics, University of Waterloo

In 1922, A. Ostrowski introduced a numeration system based on the denominators of the convergents in the continued fraction expansion of a fixed irrational number \alpha. Coquet, Rhin and Toffin studied the joint distribution in residue classes of the base-q sum-of-digits function S_q and the Ostrowski sum-of-digits function S_\alpha. They gave certain sufficient conditions for the set
{n\in\mathbb{N}: S_{q}(n)\equiv a_1\pmod{m_1},\ S_{\alpha}(n)\equiv a_2\pmod{m_2}\}
to have asymptotic density 1/m₁m₂.  In this talk, we present a quantitative version of their result when
\alpha=[0;\overline{1,m}],\ m\geq 2.

MC 5501