Dispersion properties of ergodic translations.

Isola, Stefano

Displaying similar documents to “Dispersion properties of ergodic translations.”

Constructions of smooth and analytic cocycles over irrational circle rotations

Dalibor Volný (1995)

Commentationes Mathematicae Universitatis Carolinae

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We define a class of step cocycles (which are coboundaries) for irrational rotations of the unit circle and give conditions for their approximation by smooth and real analytic coboundaries. The transfer functions of the approximating (smooth and real analytic) coboundaries are close (in the supremum norm) to the transfer functions of the original ones. This result makes it possible to construct smooth and real analytic cocycles which are ergodic, ergodic and squashable (see [Aaronson,...

The fractional dimensional theory in Lüroth expansion

Luming Shen, Kui Fang (2011)

Czechoslovak Mathematical Journal

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It is well known that every $x \in (0, 1]$ can be expanded to an infinite Lüroth series in the form of $x = \frac{1}{d_{1} (x)} + \dots + \frac{1}{d_{1} (x) (d_{1} (x) - 1) \dots d_{n - 1} (x) (d_{n - 1} (x) - 1) d_{n} (x)} + \dots,$ where $d_{n} (x) \geq 2$ for all $n \geq 1$ . In this paper, sets of points with some restrictions on the digits in Lüroth series expansions are considered. Mainly, the Hausdorff dimensions of the Cantor sets $F_{φ} = {x \in (0, 1] : d_{n} (x) \geq φ (n), \forall n \geq 1}$ are completely determined, where $φ$ is an integer-valued function defined on $ℕ$ , and $φ (n) \to \infty$ as $n \to \infty$ .

The efficiency of approximating real numbers by Lüroth expansion

Chunyun Cao, Jun Wu, Zhenliang Zhang (2013)

Czechoslovak Mathematical Journal

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For any $x \in (0, 1]$ , let $x = \frac{1}{d_{1}} + \frac{1}{d_{1} (d_{1} - 1) d_{2}} + \dots + \frac{1}{d_{1} (d_{1} - 1) \dots d_{n - 1} (d_{n - 1} - 1) d_{n}} + \dots$ be its Lüroth expansion. Denote by $P_{n} (x) / Q_{n} (x)$ the partial sum of the first $n$ terms in the above series and call it the $n$ th convergent of $x$ in the Lüroth expansion. This paper is concerned with the efficiency of approximating real numbers by their convergents ${P_{n} (x) / Q_{n} (x)}_{n \geq 1}$ in the Lüroth expansion. It is shown that almost no points can have convergents as the optimal approximation for infinitely many times in the Lüroth expansion. Consequently, Hausdorff dimension is introduced to quantify...