Linear relations among asymptotic frequencies in continued fractions

TL;DR

This paper investigates linear relations among asymptotic frequencies in continued fractions, providing bases for these relations in cases where the modulus is a prime power or a product of two distinct primes.

Contribution

It characterizes and constructs bases for the rational linear relations among frequencies in continued fractions for specific classes of moduli.

Findings

Identifies bases for relations when m is a prime power.

Determines bases for relations when m=pq, with p and q primes.

Focuses on symmetric relations c_d=c_{m-2-d}.

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.