On the asymptotic behaviour of the Lebesgue measure of sum-level sets for continued fractions

Abstract

In this paper we give a detailed measure theoretical analysis ofwhat we call sum-level sets for regular continued fractionexpansions. The first main result is to settle a recent conjecture ofFiala and Kleban, which asserts that the Lebesgue measureof these level sets decays to zero, for the level tending toinfinity. The second and third main results then give precise asymptoticestimates for this decay. The proofs of these results are based on recentprogress ininfinite ergodic theory, and in particular, they give non-trivialapplications of this theory to number theory. The paper closes with a discussion of thethermodynamical significance of the obtained results, and with someapplications of these to metrical Diophantine analysis.