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Bounded Lüroth expansions: applying Schmidt games where infinite distortion exists

Bill Mance, Jimmy Tseng (2013)

Acta Arithmetica

We show that the set of numbers with bounded Lüroth expansions (or bounded Lüroth series) is winning and strong winning. From either winning property, it immediately follows that the set is dense, has full Hausdorff dimension, and satisfies a countable intersection property. Our result matches the well-known analogous result for bounded continued fraction expansions or, equivalently, badly approximable numbers. We note that Lüroth expansions have a countably infinite Markov partition,...

Boundedly expressible sets

Jaroslav Hančl, Jan Šustek (2009)

Czechoslovak Mathematical Journal

For a given sequence a boundedly expressible set is introduced. Three criteria concerning the Hausdorff dimension of such sets are proved.

Digits and continuants in euclidean algorithms. Ergodic versus tauberian theorems

Brigitte Vallée (2000)

Journal de théorie des nombres de Bordeaux

We obtain new results regarding the precise average-case analysis of the main quantities that intervene in algorithms of a broad Euclidean type. We develop a general framework for the analysis of such algorithms, where the average-case complexity of an algorithm is related to the analytic behaviour in the complex plane of the set of elementary transformations determined by the algorithms. The methods rely on properties of transfer operators suitably adapted from dynamical systems theory and provide...

Dimension de Hausdorff de certains fractals aléatoires

Fathi Ben Nasr (1992)

Journal de théorie des nombres de Bordeaux

On construit des ensembles de Cantor aléatoires par partages successifs de rectangles, en partant d’un carré, (le nombre de divisions de la longueur peut être différent de celui de la largeur). La construction est stationnaire : elle fait intervenir des variables aléatoires indépendantes et équidistribuées. Sur ces ensembles il existe une mesure naturelle, μ , aléatoire elle aussi. Des résultats concernant les boréliens portant μ et leur dimension de Hausdorff ont déjà été obtenus par J. Peyrière...

Dimension of countable intersections of some sets arising in expansions in non-integer bases

David Färm, Tomas Persson, Jörg Schmeling (2010)

Fundamenta Mathematicae

We consider expansions of real numbers in non-integer bases. These expansions are generated by β-shifts. We prove that some sets arising in metric number theory have the countable intersection property. This allows us to consider sets of reals that have common properties in a countable number of different (non-integer) bases. Some of the results are new even for integer bases.

Dimensions des spirales

Yves Dupain, Michel Mendès France, Claude Tricot (1983)

Bulletin de la Société Mathématique de France

Diophantine Approximations of Infinite Series and Products

Ondřej Kolouch, Lukáš Novotný (2016)

Communications in Mathematics

This survey paper presents some old and new results in Diophantine approximations. Some of these results improve Erdos' results on~irrationality. The results in irrationality, transcendence and linear independence of infinite series and infinite products are put together with idea of irrational sequences and expressible sets.

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