Dimension Bound for Badly Approximable Grids

Seonhee Lim; Nicolas De Saxcé; Uri Shapira

doi:10.1093/imrn/rnx330

Dimension Bound for Badly Approximable Grids

Seonhee Lim, Nicolas De Saxcé, Uri Shapira

Technion - Israel Institute of Technology

Research output: Contribution to journal › Article › peer-review

Abstract

We show that there exists a subset of full Lebesgue measure V ⊂ ℝⁿ such that for every ϵ > 0 there exists δ > 0 such that for any v ϵ V the dimension of the set of vectors w satisfying lim inf k→∞ k^1/n(kv - w) ≥ ϵ (where(·)denotes the distance from the nearest integer) is bounded above by n-δ. This result is obtained as a corollary of a discussion in homogeneous dynamics and the main tool in the proof is a relative version of the principle of uniqueness of measures with maximal entropy.

Original language	English
Pages (from-to)	6317-6346
Number of pages	30
Journal	International Mathematics Research Notices
Volume	2019
Issue number	20
DOIs	https://doi.org/10.1093/imrn/rnx330
State	Published - 23 Oct 2019

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1093/imrn/rnx330

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title = "Dimension Bound for Badly Approximable Grids",

abstract = "We show that there exists a subset of full Lebesgue measure V ⊂ ℝn such that for every ϵ > 0 there exists δ > 0 such that for any v ϵ V the dimension of the set of vectors w satisfying lim inf k→∞ k1/n(kv - w) ≥ ϵ (where(·)denotes the distance from the nearest integer) is bounded above by n-δ. This result is obtained as a corollary of a discussion in homogeneous dynamics and the main tool in the proof is a relative version of the principle of uniqueness of measures with maximal entropy.",

author = "Seonhee Lim and \{De Saxc{\'e}\}, Nicolas and Uri Shapira",

note = "Funding Information: We would like to thank Manfred Einsiedler and Elon Lindenstrauss for valuable discussions. Seonhee Lim acknowledges the support of the Samsung Science and Technology Foundation under Project No. SSTF-BA1601-03. Nicolas de Saxc{\'e} acknowledges the warm hospitality of the mathematics department at the Technion. Uri Shapira acknowledges the support of ISF grant 357/13.",

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doi = "10.1093/imrn/rnx330",

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AU - Lim, Seonhee

AU - De Saxcé, Nicolas

AU - Shapira, Uri

N1 - Funding Information: We would like to thank Manfred Einsiedler and Elon Lindenstrauss for valuable discussions. Seonhee Lim acknowledges the support of the Samsung Science and Technology Foundation under Project No. SSTF-BA1601-03. Nicolas de Saxcé acknowledges the warm hospitality of the mathematics department at the Technion. Uri Shapira acknowledges the support of ISF grant 357/13.

PY - 2019/10/23

Y1 - 2019/10/23

N2 - We show that there exists a subset of full Lebesgue measure V ⊂ ℝn such that for every ϵ > 0 there exists δ > 0 such that for any v ϵ V the dimension of the set of vectors w satisfying lim inf k→∞ k1/n(kv - w) ≥ ϵ (where(·)denotes the distance from the nearest integer) is bounded above by n-δ. This result is obtained as a corollary of a discussion in homogeneous dynamics and the main tool in the proof is a relative version of the principle of uniqueness of measures with maximal entropy.

AB - We show that there exists a subset of full Lebesgue measure V ⊂ ℝn such that for every ϵ > 0 there exists δ > 0 such that for any v ϵ V the dimension of the set of vectors w satisfying lim inf k→∞ k1/n(kv - w) ≥ ϵ (where(·)denotes the distance from the nearest integer) is bounded above by n-δ. This result is obtained as a corollary of a discussion in homogeneous dynamics and the main tool in the proof is a relative version of the principle of uniqueness of measures with maximal entropy.

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U2 - 10.1093/imrn/rnx330

DO - 10.1093/imrn/rnx330

M3 - مقالة

SN - 1073-7928

VL - 2019

SP - 6317

EP - 6346

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 20

ER -

Dimension Bound for Badly Approximable Grids

Abstract

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