Linear relations among asymptotic frequencies in continued fractions (Q2287881)

The present article deals with the (regular) continued fraction expansion of numbers from \([0,1)\), i.e., \([0,1)\ni x=[0,a_1,a_2,a_2,\dots ]\). The attention is given to limits \[ \lim_{n\to\infty}{\frac{|\{j: j\le n, a_j \equiv d \mod m\}|}{n}}=H(m,d), \] where \(m\ge 2\) and \( d\le m\) are positive integers. In this paper, homogeneous \(\mathbb Q\)-linear relations, i.e., rational numbers \(c_d\) such that \[ \sum^{m} _{d=1}{c_dH(m,d)}=0 \] are investigated under the conditions \(c_m=c_{m-1}=c_{m-2}=0\) and \(c_d=c_{m-2-d}\) for \(1\le d \le m-3\). In addition, a basis of \(\mathbb Q\)-vector space of these relations, is studied for the cases when \(m=p^n\) is a prime power and when \(m=pq\) (including the case of \(m=2p\)), where \(p\ne q \) are primes. Some auxiliary definitions, survey, and proofs are given.