WINNING PROPERTY OF BADLY APPROXIMABLE POINTS ON CURVES

Victor Beresnevich; Erez Nesharim; Lei Yang

doi:10.1215/00127094-2022-0038

WINNING PROPERTY OF BADLY APPROXIMABLE POINTS ON CURVES

Victor Beresnevich
, Erez Nesharim
, Lei Yang

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

Abstract

We prove that badly approximable points on any analytic nondegenerate curve in Rⁿ are an absolute winning set. This confirms a key conjecture in the area stated by Badziahin and Velani in 2014 that represents a far-reaching generalization of Davenport’s problem from the 1960s. Among various consequences of our main result is a solution to Bugeaud’s problem on real numbers badly approximable by algebraic numbers of arbitrary degree. The proof relies on new ideas from fractal geometry and homogeneous dynamics.

Original language	English
Pages (from-to)	2841-2880
Number of pages	40
Journal	Duke Mathematical Journal
Volume	171
Issue number	14
DOIs	https://doi.org/10.1215/00127094-2022-0038
State	Published - 2022
Externally published	Yes

ASJC Scopus subject areas

General Mathematics

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Cite this

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title = "WINNING PROPERTY OF BADLY APPROXIMABLE POINTS ON CURVES",

abstract = "We prove that badly approximable points on any analytic nondegenerate curve in Rn are an absolute winning set. This confirms a key conjecture in the area stated by Badziahin and Velani in 2014 that represents a far-reaching generalization of Davenport{\textquoteright}s problem from the 1960s. Among various consequences of our main result is a solution to Bugeaud{\textquoteright}s problem on real numbers badly approximable by algebraic numbers of arbitrary degree. The proof relies on new ideas from fractal geometry and homogeneous dynamics.",

author = "Victor Beresnevich and Erez Nesharim and Lei Yang",

year = "2022",

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AU - Yang, Lei

PY - 2022

Y1 - 2022

N2 - We prove that badly approximable points on any analytic nondegenerate curve in Rn are an absolute winning set. This confirms a key conjecture in the area stated by Badziahin and Velani in 2014 that represents a far-reaching generalization of Davenport’s problem from the 1960s. Among various consequences of our main result is a solution to Bugeaud’s problem on real numbers badly approximable by algebraic numbers of arbitrary degree. The proof relies on new ideas from fractal geometry and homogeneous dynamics.

AB - We prove that badly approximable points on any analytic nondegenerate curve in Rn are an absolute winning set. This confirms a key conjecture in the area stated by Badziahin and Velani in 2014 that represents a far-reaching generalization of Davenport’s problem from the 1960s. Among various consequences of our main result is a solution to Bugeaud’s problem on real numbers badly approximable by algebraic numbers of arbitrary degree. The proof relies on new ideas from fractal geometry and homogeneous dynamics.

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U2 - 10.1215/00127094-2022-0038

DO - 10.1215/00127094-2022-0038

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VL - 171

SP - 2841

EP - 2880

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

IS - 14

ER -

WINNING PROPERTY OF BADLY APPROXIMABLE POINTS ON CURVES

Abstract

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