Denominators Of Convergents Research Articles

Linear recurrence sequences and periodicity of multidimensional continued fractions

Multidimensional continued fractions generalize classical continued fractions with the aim of providing periodic representations of algebraic irrationalities by means of integer sequences. We provide a characterization for periodicity of Jacobi–Perron algorithm by means of linear recurrence sequences. In particular, we prove that partial quotients of a multidimensional continued fraction are periodic if and only if numerators and denominators of convergents are linear recurrence sequences, generalizing similar results that hold for classical continued fractions.

Two theorems concerning rational approximations

The two theorems of the title constitute the mathematical results underlying well-formed scale theory. This paper includes the purely mathematical portion of a manuscript from 1988, which the authors cited the following year in N. Carey and D. Clampitt [Aspects of well-formed scales, Music Theory Spectrum 11 (1989), pp. 187–206]. Both theorems concern finite sets of the fractional parts of multiples of a positive real number θ. Such sets may be taken to represent musical scales generated by a single interval, and they have nice properties when their cardinalities are the denominators of convergents or semi-convergents (intermediate convergents) in the continued fraction representation of θ. Here, the results and their proofs are brought together in a notationally consistent framework.

A Remark on the Growth of the Denominators of Convergents

For ${\rm log}\frac{1+\sqrt{5}}{2}\leq \lambda_\ast \leq \lambda^\ast < \infty$ , let E(λ*, λ*) be the set $$ \left\{x\in [0,1):\ \mathop{\lim\inf}_{n \rightarrow \infty}\frac{\log q_n(x)}{n}=\lambda_{\ast}, \mathop{\lim\sup}_{n \rightarrow \infty}\frac{\log q_n(x)}{n}=\lambda^{\ast}\right\}.$$ It has been proved in [1] and [3] that E(λ*, λ*) is an uncountable set. In the present paper, we strengthen this result by showing that $$\dim E(\lambda_{\ast}, \lambda^{\ast}) \ge \frac{\lambda_{\ast} -\log \frac{1+\sqrt{5}}{2}}{2\lambda^{\ast}}$$ where dim denotes the Hausdorff dimension.

On the Growth of the Denominators of Convergents

Let log $$\log \left( {\left( {1 + \sqrt 5 } \right)/2} \right) \underline{\underline < } X\underline{\underline < } Y$$ . We prove that there exist non-denumerably many pairwise not equivalent irrational numbers α such that $$\mathop {\underline {\lim } }\limits_{n \to \infty } (1/n) log q_n (\alpha ) = X$$ and $$\overline {\mathop {\lim }\limits_{n \to \infty } } (1/n) log q_n (\alpha ) = Y$$ where qn(α) denotes the denominator of the nth convergent of α.

Sur le d�veloppement en fractions continues a quotients partiels impairs

Let (B n ) n be the sequence of denominators of convergents, given by the continued fraction expansion with odd partial quotients of an irrational number. For integersm>2 andk, 0≤k<m, we give (almost everywhere) the density of the set of integersn such thatB n is congruent tok modm. This result comes from an ergodic system.

?-Limit sets of smooth cylindrical cascades

Let f(x) be a smooth function on the circle S1, x mod 1,\(\smallint _{S^1 } f(x)dx = 0\), α be an irrational number, and qn be the denominators of convergents of continued fractions. In this note a classification of ω-limit sets for the cylindrical cascade $$T:(x,y) \to (x + \alpha , y + f(x)),$$ x e S1, y e R, is obtained. Criteria for the solvability of the equation g(x +α) — g(x)=f (x) are found. Estimates for the speed of decrease of the function $$h_{q_n } (x) = \sum _{i = 0}^{q_n - 1} f(x + i\alpha )$$ as n → ∞ are obtained.

Denominator Sequences of continued fractions I

Let α be an irrational number with simple continued fraction α = (a0, a1, a2,…). The problem studied is that of whether the sequence (qn) of denominators of the convergents pn/qn to α has a subsequence (Bn) = (qin} which is the sequence of denominators of convergents An/Bn to a different number α′. In other words, does there exist a subsequence {qin} which satisfies qio= 1 and For example, the sequence of denominators of convergents to ½(3 − e) is a subsequence of the sequence of denominators of convergents to ε.