An S-adic expansion of an infinite word is a way of writing it as the limit of an infinite product of substitutions (i.e., morphisms of a free monoid). Such a description is related to continued fraction expansions of numbers and vectors. Indeed, with a word is naturally associated, whenever it exists, the vector of frequencies of its letters. A fundamental example of this relation is between S...

This paper is an intensive study of the convergence of the Rogers-Ramanujan continued fraction. Let the continued fraction expansion of any irrational number t ∈ (0, 1) be denoted by [0, a1(t), a2(t), · · · ] and let the i-th convergent of this continued fraction expansion be denoted by ci(t)/di(t). Let S = {t ∈ (0, 1) : ai+1(t) ≥ φi infinitely often}, where φ = ( √ 5+1)/2. Let YS = {exp(2πit) ...

The link between automaticity and algebraicity is well established concerning power series in finite characteristics, decimal expansion continued fraction of real numbers. But the question whether fractions (objects $\mathbb

We are interested in the statistical behavior of a certain continued fraction whose associated dynamical system has infinite invariant measure. show growth rate denominator $Q_n$ $n$-th convergent negative expansion $x$ and approximation: $$ \frac{\log{n}}{n}\log{\left|x-\frac{P_n}{Q_n}\right|}\rightarrow -\frac{\pi^2}{3} \quad \text{in measure} for $x$. In course proof, we reprove related resu...

We complete the task, begun in [19], of determining when the central norm (determined by the infrastructure of the underlying real quadratic field) is equal to 2 in the simple continued fraction expansion of the associated quadratic surd.

The elementary nature and simplicity of the theory of continued fractions is mostly well disguised in the literature. This makes one reluctant to quote sources when making a remark on the subject and seems to necessitate redeveloping the theory ab initio. That had best be done succinctly. That is done here and allows the retrieval of some amusing results on pattern in the period of the continue...

We define an infinite stochastic state machine, the Battery– Discharge–Model (BDM), which simulates the behaviour of linear and jump complexity of the continued fraction expansion of multidimensional formal power series, a relevant security measure in the cryptanalysis of stream ciphers. We also obtain finite approximations to the infinite BDM, where polynomially many states suffice to approxim...