Abstract. Using techniques introduced by D. Mayer, we prove an extension of the classical Gauss–Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of successive convergents. This result has an application to the Mixmaster Universe model in general relativity. We then study some averages involving modular symbols and show that Dirichlet series related to modular forms of weight 2 can be obtained by integrating certain functions on real axis defined in terms of continued fractions. We argue that the quotient PGL(2,Z) \ P 1 (R) should be considered as non–commutative modular curve, and show that the modular complex can be seen as a sequence of K0–groups of the related crossed–product C ∗ –algebras. §0. Introduction and summary In this paper we study the interrelation between several topics: a generalization of the classical Gauss problem on the distribution of continued fractions, certain averages of modular symbols, the properties of geodesics on modular curves, the Mixmaster Universe model in general relativity, and the non–commutative geometry

We develop a general framework for the analysis of algorithms of a broad Euclidean type. The average-case complexity of an algorithm is seen to be related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithm. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory. As a consequence, we obtain precise average-case analyses of algorithms for evaluating the Jacobi symbol of computational number theory fame, thereby solving conjectures of Bach and Shallit. These methods also provide a unifying framework for the analysis of an entire class of gcd-like algorithms together with new results regarding the probable behaviour of their cost functions. 1

We obtain new results regarding the precise average-- case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the average-case complexity of an algorithm is related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide a unifying framework for the analysis of the main parameters ---digits and continuants--- that intervene in an entire class of gcd-like algorithms. We operate a general transfer from the continuous case (Continued Fraction Algorithms) to the discrete case (Euclidean Algorithms), where Ergodic Theorems are replaced by Tauberian Theorems.

This paper consists of variations upon a theme, that of limiting modular symbols introduced in [16]. We show that, on level sets of the Lyapunov exponent for the shift map T of the continued fraction expansion, the limiting modular symbol can be computed as a Birkhoff average. We show that the limiting modular symbols vanish

this paper comes from an interesting problem by Bill Gosper cited in [6, page 363, Ex. 39]. If a baseball player's batting average is :334, what is the fewest possible number of times he has been at bat? The problem, in a slightly more general setting is (see [3]): Given an interval, find in it the rational number with the smallest numerator and denominator. Gosper's solution is the following: "Express the endpoints as continued fractions. Find the first term where they differ and add 1 to the lesser term, unless it's last. Discard the terms to the right. What's left is the continued fraction for the smallest rational 2 in the interval. (If one fraction terminates but matches the other as far as it goes, append an infinity and proceed as above.)" This problem gave us the ideas: what if we reverse the question? Given a rational number, P=Q,whatistheset of real numbers for which P=Q is a `best approximation', either of the first kind or the second? Is it an interval or a more complicated set? In the case of best appoximations of the first kind, it seems quite natural that this set is an interval. But in the case of the best approximations of the second kind the remark that follows their definition makes it not so obvious. We call these sets Optimality Intervals and the purpose of this paper is to prove that they are intervals indeed. More formally:

Abstract. This paper continues the study of the structures induced on the “invisible boundary ” of the modular tower and extends some results of [MaMar1]. We start with a systematic formalism of pseudo–measures generalizing the well– known theory of modular symbols for SL(2). These pseudo–measures, and the related integral formula which we call the Lévy–Mellin transform, can be considered as an “∞–adic ” version of Mazur’s p–adic measures that have been introduced in the seventies in the theory of p–adic interpolation of the Mellin transforms of cusp forms, cf. [Ma2]. A formalism of iterated Lévy–Mellin transform in the style of [Ma3] is sketched. Finally, we discuss the invisible boundary from the perspective of non–commutative geometry. When the theory of modular symbols for the SL(2)–case had been conceived in the 70’s (cf. [Ma1], [Ma2], [Sh1], [Sh2]), it was clear from the outset that it dealt with the Betti homology of some basic moduli spaces (modular curves, Kuga varieties, M1,n, and alike), whereas the theory of modular forms involved the de