On sums of powers of inverse complete quotients

Abstract:

For an irrational number $x$, let $x_n$ denote its $n$-th continued fraction inverse complete quotient, obtained by deleting the first $n$ partial quotients. For any positive real number $r$, we establish the optimal linear bound on the sum of the $r$-th powers of the first $n$ complete quotients. That is, we find the smallest constants $\alpha (r), \beta (r)$ such that $x_1^r+\ldots +x_n^r< \alpha (r)n+\beta (r)$ for all $n\ge 1$ and all irrationals $x$.