Rational complexity of binary sequences, F$\mathbb {Q}$SRs, and pseudo-ultrametric continued fractions in $\mathbb {R}$

Continued Fractions and Heavy Sequences

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...

Pseudo-factorials, Elliptic Functions, and Continued Fractions

This study presents miscellaneous properties of pseudo-factorials, which are numbers whose recurrence relation is a twisted form of that of usual factorials. These numbers are associated with special elliptic functions, most notably, a Dixonian and a Weierstraß function, which parametrize the Fermat cubic curve and are relative to a hexagonal lattice. A continued fraction expansion of the ordin...

Orthogonal Rational Functions and Continued Fractions

Notes on Continued Fractions and Recurrence Sequences

Introduction. The purpose of these notes is to support the papers included in this volume by providing self-contained summaries of related mathematics emphasising those aspects actually used or required. I have selected two topics that are easily described from first principles yet for which it is peculiarly difficult to find congenial introductions that warrant citing.

Hyperquadratic continued fractions and automatic sequences

We show that three different families of hyperquadratic elements, studied in the literature, have the following property: For these elements, the leading coefficients of the partial quotients in their continued fraction expansion form 2-automatic sequences. We also show that this is not true for algebraic elements in F(q) in general. Indeed, we use an element studied by Mills and Robbins as cou...