An effective asymptotic result for the Lebesgue measure of the sum-level sets for continued fractions

Byron Heersink

Published 2015-03-25Version 1

For every positive integer n, let C_n be the set of real numbers in [0,1] whose continued fraction expansion [a_1,a_2,...] satisfies a_1+...+a_k=n for some k. Using techniques from infinite ergodic theory, Kessebohmer and Stratmann proved that the Lebesgue measure of C_n is asymptotically equivalent to 1/log_2(n) as n approaches infinity. In this paper, we provide an error term for this result, employing mostly basic properties of the transfer operator of the Farey map and an adaptation of Freud's effective version of Karamata's Tauberian theorem.