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Non-Expanding Random walks on Homogeneous spaces and Diophantine approximation

Gaurav Aggarwal, Anish Ghosh

TL;DR

This work develops a comprehensive framework for non-expanding random walks on the affine-lattice space and establishes a classification of stationary measures when the projection to the unimodular lattice space is fixed. Central to the approach is an exponential-drift mechanism that compensates for contracting directions, enabling a decomposition into homogeneous components and a complete classification of stationary measures. The authors then connect these dynamical results to homogeneous dynamics via a Random Genericity principle, showing that random equidistribution and Birkhoff genericity are equivalent in this setting. Finally, they apply these results to inhomogeneous Diophantine approximation on fractals, proving zero-measure results for badly approximable and Dirichlet-improvable sets for broad classes of fractal measures (Type 1–3), including new singly metric cases on the middle-third Cantor set and related fractals.

Abstract

We study non-expanding random walks on the space of affine lattices and establish a new classification theorem for stationary measures. Further, we prove a theorem that relates the genericity with respect to these random walks to Birkhoff genericity. Finally, we apply these theorems to obtain several results in inhomogeneous Diophantine approximation, especially on fractals.

Non-Expanding Random walks on Homogeneous spaces and Diophantine approximation

TL;DR

This work develops a comprehensive framework for non-expanding random walks on the affine-lattice space and establishes a classification of stationary measures when the projection to the unimodular lattice space is fixed. Central to the approach is an exponential-drift mechanism that compensates for contracting directions, enabling a decomposition into homogeneous components and a complete classification of stationary measures. The authors then connect these dynamical results to homogeneous dynamics via a Random Genericity principle, showing that random equidistribution and Birkhoff genericity are equivalent in this setting. Finally, they apply these results to inhomogeneous Diophantine approximation on fractals, proving zero-measure results for badly approximable and Dirichlet-improvable sets for broad classes of fractal measures (Type 1–3), including new singly metric cases on the middle-third Cantor set and related fractals.

Abstract

We study non-expanding random walks on the space of affine lattices and establish a new classification theorem for stationary measures. Further, we prove a theorem that relates the genericity with respect to these random walks to Birkhoff genericity. Finally, we apply these theorems to obtain several results in inhomogeneous Diophantine approximation, especially on fractals.
Paper Structure (19 sections, 29 theorems, 163 equations)

This paper contains 19 sections, 29 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Random Walks
  4. Notation
  5. Basic Facts and Observations
  6. Conditional Measures
  7. Horocycle Flows
  8. The exponential drift
  9. Disintegration of $\nu_b$ along the stabilizers
  10. Proof of Theorem \ref{['Main Random walk']}
  11. Random Genericity: Proof of Theorem \ref{['Random Genericity']}
  12. Application to Homogeneous Dynamics
  13. Type 1 measures
  14. Proof of Theorem \ref{['Intermediate Theorem']}
  15. Diophantine Approximation
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\nu$ be a measure on $G$ as above. Then any probability measure $\mu$ on $\mathcal{X}$ which is $\nu$-stationary and satisfies $\pi_* \mu= \mu_{\mathcal{X}'}$, must equal $\mu_{\mathcal{X}}$.

Theorems & Definitions (72)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.11
  • ...and 62 more