On certain statistical properties of continued fractions with even and with odd partial quotients

Abstract

We prove results concerning the joint limiting distribution of the renewal time of denominators and consecutive digits of random irrational numbers in the case of continued fractions with even partial quotients, with odd partial quotients, and for the nearest integer continued fraction. Let (qn) denote the sequence of denominators of successive convergents in the regular con- tinued fraction (RCF) expansion of an irrational number. For each R > 1, consider the renewal time nR := min{n : qn > R}, so that qnR−1 6 R < qnR . As a consequence of their renewal-type theorem for the natural extension of the Gauss map associated with regular continued fractions (RCF), Sinai and Ulcigrai proved the existence of the joint limiting dis- tribution of q nR−1 R , R qnR , anR−K, . . . , anR+K � with K a fixed nonnegative integer (SU). This result has been subsequently extended to the situation of continued fractions with even par- tial quotients (ECF) by Cellarosi (C1). This ECF limiting distribution was further used in the renormalization of theta sums, leading to some new results about the distribution of normalized theta sums and geometrical properties of their associated curlicues (C2, Si). Em- ploying an abstract characterization of denominators of successive convergents in the regular continued fraction expansion RCF(x) of x, Ustinov succeeded in explicitly computing the limiting distribution in the RCF case (U1). This paper considers the corresponding problem in the ECF case and in the case of continued fractions with odd partial quotients (OCF). Its purposes are to simplify the proof of the main result in (C1) while making the limiting distribution explicit, and to extend this type of results to OCF, for which no ergodic theoretical approach is known at this time. As in (U1), the key tool is providing an abstract characterization for pairs of successive convergents in ECF(x) and OCF(x), which may be of independent interest. The OCF case is more complicated because the sequence of denominators of successive convergents in OCF(x) is not necessarily increasing, as in the RCF or ECF cases. We next review some basic properties of ECF (respectively, OCF expansions), given for each x ∈ := (0,1) Q by