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Winning and nullity of inhomogeneous bad

Shreyasi Datta, Liyang Shao

TL;DR

This work establishes that inhomogeneous weighted badly approximable vectors form a Hyperplane Absolute Winning set, extending the BNY21 framework to inhomogeneous data through a Cantor-game strategy that localizes the inhomogeneous function and leverages α-Ahlfors regular absolutely decaying measures. It then derives a measure-zero nullity result: for any nondegenerate curve or nondegenerate analytic manifold, almost every point is not in $Bad_theta(w)$ for any weight, achieved via duality and quantitative nondivergence from homogeneous dynamics. The methods combine Cantor-game techniques with dual-lattice and nondivergence controls to handle inhomogeneous constraints and higher-dimensional manifolds, highlighting a robust largeness-nullity dichotomy and suggesting avenues for dual-analytic extensions in Diophantine approximation.

Abstract

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 Analysis, 31 (1), 1-33, 2021] to the inhomogeneous set-up. We also show for any nondegenerate curve and nondegenerate analytic manifold that almost every point is not weighted inhomogeneous badly approximable for any weight. This is achieved by duality and the quantitative nondivergence estimates from homogeneous dynamics motivated by [Acta Math. 231 (2023), 1-30], together with the methods from [arXiv:2307.10109].

Winning and nullity of inhomogeneous bad

TL;DR

This work establishes that inhomogeneous weighted badly approximable vectors form a Hyperplane Absolute Winning set, extending the BNY21 framework to inhomogeneous data through a Cantor-game strategy that localizes the inhomogeneous function and leverages α-Ahlfors regular absolutely decaying measures. It then derives a measure-zero nullity result: for any nondegenerate curve or nondegenerate analytic manifold, almost every point is not in for any weight, achieved via duality and quantitative nondivergence from homogeneous dynamics. The methods combine Cantor-game techniques with dual-lattice and nondivergence controls to handle inhomogeneous constraints and higher-dimensional manifolds, highlighting a robust largeness-nullity dichotomy and suggesting avenues for dual-analytic extensions in Diophantine approximation.

Abstract

We prove the hyperplane absolute winning property of weighted inhomogeneous badly approximable vectors in . This answers a question by Beresnevich--Nesharim--Yang and extends the main result of [Geometric and Functional Analysis, 31 (1), 1-33, 2021] to the inhomogeneous set-up. We also show for any nondegenerate curve and nondegenerate analytic manifold that almost every point is not weighted inhomogeneous badly approximable for any weight. This is achieved by duality and the quantitative nondivergence estimates from homogeneous dynamics motivated by [Acta Math. 231 (2023), 1-30], together with the methods from [arXiv:2307.10109].

Paper Structure

This paper contains 17 sections, 26 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Hyperplane absolute winning
  3. Inhomogeneous Bad is null
  4. Preliminaries
  5. Preliminaries for Theorem \ref{['thm: main']}
  6. Preliminaries for Theorem \ref{['measure-zero-theorem']}
  7. Hyperplane absolute winning: Theorem \ref{['thm: main']}
  8. Preparation for the proof of Theorem \ref{['thm: main']}
  9. Recalling the game in BNY21
  10. The new strategy
  11. Proof of Theorem \ref{['thm: main']} modulo Lemma \ref{['cantorwinning']}
  12. Proof of Lemma \ref{['cantorwinning']}
  13. Finishing the proof of Lemma \ref{['cantorwinning']}
  14. Nulity of weighted inhomogeneous bad in a manifold: Theorem \ref{['measure-zero-theorem']}
  15. Finishing the proof of Theorem \ref{['measure-zero-theorem']}
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\mathbf{\boldsymbol\theta}:\mathbb{R}^d\to\mathbb{R}^d$, with $\mathbf{\boldsymbol\theta}(\mathbf{x}):=(\theta_i(x_i))$, $\theta_i:\mathbb{R}\to \mathbb{R}$ Lipschitz function and $\mathbf{w}$ be a weight in $\mathbb{R}^d$. Then $\mathbf{Bad}_{\mathbf{\boldsymbol\theta}}(\mathbf{w})$ is hyperpl

Theorems & Definitions (45)

  • Theorem 1
  • Corollary 1.1
  • Theorem 2
  • Corollary 1.2
  • Remark 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 3
  • ...and 35 more