  In this paper we give a detailed measure theoretical analysis of what we call
sum-level sets for regular continued fraction expansions. The first main result
is to settle a recent conjecture of Fiala and Kleban, which asserts that the
Lebesgue measure of these level sets decays to zero, for the level tending to
infinity. The second and third main result then give precise asymptotic
estimates for this decay. The proofs of these results are based on recent
progress in infinite ergodic theory, and in particular, they give non-trivial
applications of this theory to number theory. The paper closes with a
discussion of the thermodynamical significance of the obtained results, and
with some applications of these to metrical Diophantine analysis.
