Schmidt's Game Research Articles

Badly approximable grids and k$k$‐divergent lattices

AbstractLet be a matrix. In this paper, we investigate the set of badly approximable targets for , where is the ‐torus. It is well known that is a winning set for Schmidt's game and hence is a dense subset of full Hausdorff dimension. We investigate the relationship between the measure of and Diophantine properties of . On the one hand, we give the first examples of a nonsingular such that has full measure with respect to some nontrivial algebraic measure on the torus. For this, we use transference theorems due to Jarnik and Khintchine, and the parametric geometry of numbers in the sense of Roy. On the other hand, we give a novel Diophantine condition on that slightly strengthens nonsingularity, and show that under the assumption that satisfies this condition, is a null‐set with respect to any nontrivial algebraic measure on the torus. For this, we use naive homogeneous dynamics, harmonic analysis, and a novel concept that we refer to as mixing convergence of measures.

A variational principle in the parametric geometry of numbers

We extend the parametric geometry of numbers (initiated by Schmidt and Summerer, and deepened by Roy) to Diophantine approximation for systems of m linear forms in n variables, and establish a new connection to the metric theory via a variational principle that computes fractal dimensions of a variety of sets of number-theoretic interest. The proof of our variational principle relies on two novel ingredients: a variant of Schmidt's game capable of computing the Hausdorff and packing dimensions of any set, and the notion of templates, which generalize Roy's rigid systems. We use our variational principle to compute the Hausdorff and packing dimensions of the set of singular systems of linear forms and show they are equal, resolving a conjecture of Kadyrov, Kleinbock, Lindenstrauss and Margulis, as well as a question of Bugeaud, Cheung and Chevallier. As a corollary of Dani's correspondence principle, the divergent trajectories of a one-parameter diagonal action on the space of unimodular lattices with exactly two Lyapunov exponents with opposite signs have equal Hausdorff and packing dimensions. Other applications include quantitative strengthenings of theorems due to Cheung and Moshchevitin, which originally resolved conjectures due to Starkov and Schmidt respectively; as well as dimension formulas with respect to the uniform exponent of irrationality for simultaneous and dual approximation in two dimensions, completing partial results due to Baker, Bugeaud, Cheung, Chevallier, Dodson, Laurent and Rynne.

Quantitative results using variants of Schmidt’s game: Dimension bounds, arithmetic progressions, and more

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 paper we show that such quantitative information has applications to various questions including: * What is the maximal length of an arithmetic progression on the middle $\epsilon$ Cantor set? * What is the smallest $n$ such that there is some element of the ternary Cantor set whose continued fraction partial quotients are all $\leq n$? * What is the Hausdorff dimension of the set of $\epsilon$-badly approximable numbers on the Cantor set? We show that a variant of Schmidt's game known as the $potential$ $game$ is capable of providing better bounds on the answers to these questions than the classical Schmidt's game. We also use the potential game to provide a new proof of an important lemma in the classical proof of the existence of Hall's Ray.

Decaying and non-decaying badly approximable numbers

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 the set of decaying badly approximable numbers, and also of the set of badly approximable numbers which are not decaying. We answer both questions, showing that the Hausdorff dimensions of both sets are equal to one. Part of our proof utilizes a game which combines the Banach--Mazur game and Schmidt's game, first introduced in Fishman, Reams, and Simmons (preprint '15).

The Banach–Mazur–Schmidt and Banach–Mazur–McMullen games

We introduce two new mathematical games, the Banach–Mazur–Schmidt game and the Banach–Mazur–McMullen game, merging well-known games. We investigate the properties of the games, as well as providing an application to Diophantine approximation theory, analyzing the geometric structure of certain Diophantine sets.

2-dimensional badly approximable vectors and Schmidt’s game

We prove that for any pair $(s,t)$ of nonnegative numbers with $s+t=1$, the set of two-dimensional $(s,t)$-badly approximable vectors is winning for Schmidt's game. As a consequence, we give a direct proof of Schmidt's conjecture using his game.

On Schmidt's game and the set of points with non-dense orbits under a class of expanding maps

Given an expanding interval map, we consider sets of points whose forward orbits are bounded away from a given point. It is shown that such sets are 1/2-winning in the sense of Schmidt's game for a class of expanding interval maps including the Gauss transformation and all beta transformations. This improves upon several results dating back to a work of W. Schmidt.

Bad(s,t) is hyperplane absolute winning

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

Badziahin-Pollington-Velani's theorem and Schmidt's game

We prove that for any s, t ⩾ 0 with s + t = 1 and any θ ∈ ℝ with infq∈ℕq1/s||qθ|| > 0, the set of y ∈ ℝ for which (θ, y) is (s, t)-badly approximable is ½-winning for Schmidt's game. As a consequence, we remove a technical assumption in a recent theorem of Badziahin–Pollington–Velani on simultaneous Diophantine approximation.

Winning games for bounded geodesics in moduli spaces of quadratic differentials

We prove that the set of bounded geodesics in Teichmuller space is a winning set for Schmidt's game. This is a notion of largeness in a metric space that can apply to measure $0$ and meager sets. We prove analogous closely related results on any Riemann surface, in any stratum of quadratic differentials, on any Teichmuller disk and for intervals exchanges with any fixed irreducible permutation.

Schmidt's game, badly approximable matrices and fractals

We prove that for every M , N ∈ N , if τ is a Borel, finite, absolutely friendly measure supported on a compact subset K of R M N , then K ∩ BA ( M , N ) is a winning set in Schmidt's game sense played on K , where BA ( M , N ) is the set of badly approximable M × N matrices. As an immediate consequence we have the following application. If K is the attractor of an irreducible finite family of contracting similarity maps of R M N satisfying the open set condition (the Cantor's ternary set, Koch's curve and Sierpinski's gasket to name a few known examples), then dim K = dim K ∩ BA ( M , N ) .

Frequencies of digits, divergence points, and Schmidt games

Sets of divergence points, i.e. numbers x (or tuples of numbers) for which the limiting frequency of a given string of N-adic digits of x fails to exist, have recently attracted huge interest in the literature. In this paper we consider sets of simultaneous divergence points, i.e. numbers x (or tuples of numbers) for which the limiting frequencies of all strings of N-adic digits of x fail to exist. We show that many natural sets of simultaneous divergence points are ( α, β)-wining sets in the sense of the Schmidt game. As an application we obtain lower bounds for the Hausdorff dimension of these sets.

Sublacunary Sequences and Winning Sets

Recall the definition of an (α, β)-Banach–Mazur–Schmidt game on the real line, an (α, β)and α-winning set, and their simplest properties (see [1, 2]). Suppose that 0 0 . Schmidt [1, 2] proved that the intersection of countably many α-winning sets is again an α-winning set. As an example of an α-winning set with an arbitrary α ∈ (0, 1/2) , Schmidt considered the set

On orbits of endomorphisms of tori and the Schmidt game

AbstractWe show that there exists a subset F of the n-dimensional torus n such that F has Hausdorff dimension n and for any x∈F and any semisimple automorphism σ of n the closure of the σ-orbit of x contains no periodic points.