Linear relations among asymptotic frequencies in continued fractions

Abstract

Let $H(m,d)$ denote the asymptotic frequency of the natural numbers $k\equiv d \mod m$ in the continued fraction expansions of almost all numbers $x\in[0,1)$. For a fixed number $m\ge 4$, we study $\mathbb Q$-linear relations among the numbers $H(m,d)$, $1\le d\le m-3$, i.e., vectors $(c_1,\ldots,c_{m-3})\in\mathbb Q^{m-3}$ such that $$ \sum_{d=1}^{m-3} c_dH(m,d)=0. $$ We restrict ourselves to the symmetric case $c_d=c_{m-2-d}$. In the end, we obtain a basis of the $\mathbb Q$-vector space of these relations for prime powers $m$ and for $m=pq$, where $p\ne q$ are primes.