Related papers: Cantor-winning sets and their applications

We show that the sets of weighted badly approximable vectors in $\Bbb R^n$ are winning sets of certain games, which are modifications of $(\alpha,\beta)$-games introduced by W. Schmidt in 1966. The latter winning property is stable with…

Schmidt games and the Cantor winning property give alternative notions of largeness, similar to the more standard notions of measure and category. Being intuitive, flexible, and applicable to recent research made them an active object of…

We prove that the countable intersection of $C^1$-diffeomorphic images of certain Diophantine sets has full Hausdorff dimension. For example, we show this for the set of badly approximable vectors in $\mathbb{R}^d$, improving earlier…

We construct (\alpha ,\beta) and \alpha -winning sets in the sense of Schmidt's game, played on the support of certain measures (very friendly and awfully friendly measures) and show how to derive the Hausdorff dimension for some. In…

In \cite{SchmidtGames}, W. Schmidt proved that the set of non-normal numbers in base $b$ is a {\it winning set}. We generalize this result by proving that many sets of non-normal numbers with respect to the Cantor series expansion are…

Let (X,d) be a metric space and (\Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of \Omega. Loosely speaking, these consist of points in \Omega…

We give sufficient conditions for two Cantor sets of the line to be nested for a positive set of translation parameters. This problem occurs in diophantine approximations. It also occurs as a toy model of the parameter selection for…

We consider a natural filtration $\boldsymbol{\operatorname{Bad}}(\delta) \subset \boldsymbol{\operatorname{Bad}}(\delta')$ for $\delta \geq \delta'>0$ on the set of badly approximable numbers to complement the filtration of the well…

Winning sets of Schmidt's game enjoy a remarkable rigidity. Therefore, this game (and modifications of it) have been applied to many examples of complete metric spaces (X, d) to show that the set of "badly approximable points", with respect…

We prove that the Littlewood conjecture is satisfied for a restricted class of pairs $(\alpha,\beta)$ of badly approximable numbers. We use the localization of the roots of a cubic equation with coefficients depending on the diophantine…

Schmidt's game is generally used to deduce qualitative information about the Hausdorff dimensions of fractal sets and their intersections. However, one can also ask about quantitative versions of the properties of winning sets. In this…

The set of badly approximable numbers, Bad, is known to be winning for Schmidt's game and hence has full Hausdorff dimension. It is also known that the set of inhomogeneously badly approximable numbers has full dimension. We prove that the…

In 1984, Kurt Mahler posed the following fundamental question: How well can irrationals in the Cantor set be approximated by rationals in the Cantor set? Towards development of such a theory, we prove a Dirichlet-type theorem for this…

We present a new variant of the potential game and show that certain compact subsets of $\R^n$, including a large class of self-affine sets, are winning in our game. We prove that sets with sufficiently strong winning conditions are…

In this work we reproduce the characterization of $\Gg^s$-sets from the euclidean setting [J. London Math. Soc. 49:267-280,1994] to more general metric spaces. These sets have Hausdorff dimension at least $s$ and are closed by countable…

In this paper we study a class of random Cantor sets. We determine their almost sure Hausdorff, packing, box, and Assouad dimensions. From a topological point of view, we also compute their typical dimensions in the sense of Baire category.…

The use of Hausdorff measures and dimension in the theory of Diophantine approximation dates back to the 1920s with the theorems of Jarnik and Besicovitch regarding well-approximable and badly-approximable points. In this paper we consider…

Fundamental questions in Diophantine approximation are related to the Hausdorff dimension of sets of the form $\{x\in \mathbb{R}: \delta_x = \delta\}$, where $\delta \geq 1$ and $\delta_x$ is the Diophantine approximation rate of an…

Let $K$ denote the middle third Cantor set and ${\cal A}:= \{3^n : n = 0,1,2, >... \} $. Given a real, positive function $\psi$ let $ W_{\cal A}(\psi)$ denote the set of real numbers $x$ in the unit interval for which there exist infinitely…

In this paper we study the classical Schmidt game on two families of sets: one related to frequencies of digits in base-$2$ expansions, and one connected to the set of the badly approximable numbers. Namely, we describe some nontrivial…