Continued fractions and Dedekind sums: Three-term relations and distribution

Abstract

Let 0 ⩽ m < N be coprime integers and T ( m , N ) the sum of the digits of the continued fraction expansion of m / N . We study the distribution of large values of T ( m , N ) (“large” means T ( m , N ) ⩾ N α for some fixed exponent α when N tends to infinity), which seems to be quite similar to that of the absolute values of the corresponding Dedekind sums S ( m , N ) . This similarity is (partially, at least) due to analogous three-term relations holding for both kinds of sums. We show, e.g., | { m : 0 ⩽ m < N , ( m , N ) = 1 , T ( m , N ) ⩾ N α } | ⩾ C ( α ) ϕ ( N ) log N / N α , where the right-hand side has the best possible order of magnitude one can expect. The same type of estimate holds for | S ( m , N ) | and α > 1 / 3 ; in the case α ⩽ 1 / 3 we have to drop the factor log N .