It is widely believed that the continued fraction expansion of every irrational algebraic number α either is eventually periodic (and we know that this is the case if and only if α is a quadratic irrational), or it contains arbitrarily large partial quotients. Apparently, this question was first considered by Khintchine in [22] (see also [6,39,41] for surveys including a discussion on this subj...

We find a generating function expressed as a continued fraction that enumerates ordered trees by the number of vertices at different levels. Several Catalan problems are mapped to an ordered-tree problem and their generating functions also expressed as a continued fraction. Among these problems is the enumeration of (132)-pattern avoiding permutations that have a given number of increasing patt...

Babson and Steingrimsson (2000, Séminaire Lotharingien de Combinatoire, B44b, 18) introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Let fτ ;r (n) be the number of 1-3-2-avoiding permutations on n letters that contain exactly r occurrences of τ , where τ is a generalized pattern on k letters. Let Fτ ...

A set S of positive integers is avoidable if there exists a partition of the positive integers into two disjoint sets such that no two distinct integers from the same set sum to an element of S. Much previous work has focused on proving the avoidability of very special sets of integers. We vastly broaden the class of avoidable sets by establishing a previously unnoticed connection with the elem...

It is widely recognized that the work of Ramanujan deeply influenced the direction of modern number theory. This influence resonates clearly in the “Ramanujan conjectures.” Here I will explore another part of his work whose position within number theory seems to be less well understood, even though it is more elementary, namely that related to continued fractions. I will concentrate on the spec...

We show connections between a special type of addition formulas and a theorem of Stieltjes and Rogers. We use different techniques to derive the desirable addition formulas. We apply our approach to derive special addition theorems for Bessel functions and confluent hypergeometric functions. We also derive several addition theorems for basic hypergeometric functions. Applications to the evaluat...

This incursion into the realm of elementary and probabilistic number theory of continued fractions, via modular forms, allows us to study the alternating sum of coeecients of a continued fraction, thus solving the longstanding open problem of their limit law.

We initiate the study of the sets H(c), 0<c<1, of real x for which the sequence (kx)k≥1 (viewed mod 1) consistently hits the interval [0, c) at least as often as expected (i. e., with frequency ≥ c). More formally, H(c) def = {α ∈ R ∣ card({1 ≤ k ≤ n ∣ ⟨kα⟩ < c}) ≥ cn, for all n ≥ 1}, where ⟨x⟩ = x− [x] stands for the fractional part of x ∈ R. We prove that, for rational c, the sets H(c) are of...

We introduce the notion of “slope” for geodesic laminations. Slope is a positive irrational defined via regular continued fraction. The action of the mapping class group on lamination pulls back to the action of GL(2, Z) on real line. We discuss applications of slopes in complex analysis, low-dimensional topology, geometric group theory and C∗-algebras.