Badly Approximable Systems of Affine Forms

Abstract. We present an account of some recent applications of ergodic theorems for actions of algebraic and arithmetic groups to the solution of natural problems in Diophantine approximation and number theory. Our approach is based on spectral methods utilizing the unitary representation theory of the groups involved. This allows the derivation of ergodic theorems with a rate of convergence, an important phenomenon which does not arise in classical ergodic theory. Combining spectral and dynamical methods, quantitative ergodic theorems give rise to new and previously inaccessible applications. We demonstrate the remarkable diversity of such applications by deriving general uniform error estimates in non-Euclidean lattice points counting problems, explicit estimates in the sifting problem for almost-prime points on symmetric varieties, best-possible bounds for exponents of intrinsic Diophantine approximation on homogeneous algebraic varieties, and quantitative results on fast distribution of dense orbits on compact and non-compact homogeneous spaces.

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“…Kleinbock,and G. Margulis [D85,D86,D89,K98a,K98b,KM98,K99,KM99]. These developments are surveyed in [K01, M02, K10].…”

Section: Intrinsic Diophantine Approximation On Homogeneous Varietiesmentioning

confidence: 99%

Quantitative ergodic theorems and their number-theoretic applications

Gorodnik

Nevo

2014

Bull. Amer. Math. Soc.

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“…The element in M m,n (R) corresponding to A ∈ M m,n (R) and b ∈ R m will be expressed as A, b . Consider the following well-known sets from the theory of Diophantine approximation [8]: The set Bad 0 (m, n) is called the set of badly approximable systems of m linear forms in n variables and is an important and classical object of study in the theory of Diophantine approximation. Although it is a Lebesgue null set (Khintchine, 1926), it has full Hausdorff dimension and, even stronger, is winning (Schmidt, 1969 and Bad(m, n) are winning instead of just having full Hausdorff dimension.…”

Section: Introductionmentioning

confidence: 99%

Badly approximable affine forms and Schmidt games

Tseng

2009

Journal of Number Theory

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“…The set y ∈ π −1 (x) : µ(y) > 0 (where x ∈ X 2 is arbitrary) has also been investigated, and it is known to have full Hausdorff dimension as well as being a winning set for Schmidt's game. See for example [13], [5], [22], and [11].…”

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confidence: 99%

A solution to a problem of Cassels and Diophantine properties of cubic numbers

Shapira

2011

Ann. Math.

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We prove that almost any pair of real numbers α, β, satisfies the following inhomogeneous uniform version of Littlewood's conjecture:where · denotes the distance from the nearest integer. The existence of even a single pair that satisfies statement (C1), solves a problem of Cassels from the 50's. We then prove that if 1, α, β span a totally real cubic number field, then α, β, satisfy (C1). This generalizes a result of Cassels and Swinnerton-Dyer, which says that such pairs satisfy Littlewood's conjecture. It is further shown that if α, β are any two real numbers, such that 1, α, β, are linearly dependent over Q, they cannot satisfy (C1). The results are then applied to give examples of irregular orbit closures of the diagonal group of a new type. The results are derived from rigidity results concerning hyperbolic actions of higher rank commutative groups on homogeneous spaces.

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Badly Approximable Systems of Affine Forms

Cited by 28 publications

References 16 publications

Quantitative ergodic theorems and their number-theoretic applications

Quantitative ergodic theorems and their number-theoretic applications

Badly approximable affine forms and Schmidt games

A solution to a problem of Cassels and Diophantine properties of cubic numbers

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