Related papers: Badly approximable $S$-numbers and absolute Schmid…

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…

We prove that the Hausdorff dimension of the set of badly approximable systems of m linear forms in n variables over the field of Laurent series with coefficients from a finite field is maximal. This is a analogue of Schmidt's…

We call a badly approximable number $decaying$ if, roughly, the Lagrange constants of integer multiples of that number decay as fast as possible. In this terminology, a question of Y. Bugeaud ('15) asks to find the Hausdorff dimension of…

We extend the work of An, Guan and Kleinbock on bounded orbits of diagonalizable flows on $\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})$ to $\mathrm{SL}_3(\mathbb{C})/\mathrm{SL}_3(\mathcal{O}_{\mathbb{K}})$, where $\mathbb{K}$ is an…

We introduce and develop a class of \textit{Cantor-winning} sets that share the same amenable properties as the classical winning sets associated to Schmidt's $(\alpha,\beta)$-game: these include maximal Hausdorff dimension, invariance…

For any i,j>0 with i+j =1, let Bad(i,j) denote the set of points (x,y) \in R^2 such that max \{ ||qx||^{1/i}, \, ||qy||^{1/j} \} > c/q for some positive constant c = c(x,y) and all q in N. We show that \Bad(i,j) \cap C is winning in the…

J. An (2013) proved that for any $s,t \geq 0$ such that $s + t = 1$, $\mathbf{Bad}(s,t)$ is $(34\sqrt 2)^{-1}$-winning for Schmidt's game. We show that using the main lemma from An's paper one can derive a stronger result, namely that…

We prove that the set of (r_1,r_2,..,r_{d})-badly approximable vectors is a winning set if r_1=r_2=...=r_{d-1}\geq r_{d}.

Given a group $G$ and a number field $K$, the Grunwald problem asks whether given field extensions of completions of $K$ at finitely many places can be approximated by a single field extension of $K$ with Galois group G. This can be viewed…

We prove that for any $s,t\ge0$ with $s+t=1$ and any $\theta\in\mathbb{R}$ with $\inf_{q\in\mathbb{N}}q^{\frac{1}{s}}\|q\theta\|>0$, the set of $y\in\mathbb{R}$ for which $(\theta,y)$ is $(s,t)$-badly approximable is 1/2-winning for…

We prove the hyperplane absolute winning property of weighted inhomogeneous badly approximable vectors in $\mathbb{R}^d$. This answers a question by Beresnevich--Nesharim--Yang and extends the main result of [Geometric and Functional…

We show that the set of numbers with bounded L\"uroth expansions (or bounded L\"uroth series) is winning and strong winning. From either winning property, it immediately follows that the set is dense, has full Hausdorff dimension, and…

We consider Schmidt's game on the space of compact subsets of a given metric space equipped with the Hausdorff metric, and the space of continuous functions equipped with the supremum norm. We are interested in determining the generic…

While many types of non-measurable sets are never $(\alpha, \beta)$-winning in the sense of Schmidt's game, we show that this is not the case for certain Vitali sets. Our main theorems show that for certain values of $\alpha, \beta$ one can…

We prove the absolute winning property of weighted simultaneous inhomogeneous badly approximable vectors on non-degenerate analytic curves. This answers a question by Beresnevich, Nesharim, and Yang. In particular, our result is an…

We prove that a Hausdorff space $X$ is very $\mathrm I$-favorable if and only if $X$ is the almost limit space of a $\sigma$-complete inverse system consisting of (not necessarily Hausdorff) second countable spaces and surjective d-open…

Given $b > 1$ and $y \in \mathbb{R}/\mathbb{Z}$, we consider the set of $x\in \mathbb{R}$ such that $y$ is not a limit point of the sequence $\{b^n x \bmod 1: n\in\mathbb{N}\}$. Such sets are known to have full Hausdorff dimension, and in…

We establish a strong form of Littlewood's conjecture with inhomogeneous shifts, for a full-dimensional set of pairs of badly approximable numbers on a vertical line. We also prove a uniform assertion of this nature, generalising a strong…

The Reifenberg theorem \cite{reif_orig} tells us that if a set $S\subseteq B_2\subseteq \mathbb R^n$ is uniformly close on all points and scales to a $k$-dimensional subspace, then $S$ is H\"older homeomorphic to a $k$-dimensional Euclidean…

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…