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A shrinking target problem in homogeneous spaces of semisimple algebraic groups

Cheng Zheng

TL;DR

This paper analyzes shrinking target problems at infinity on homogeneous spaces of connected semisimple algebraic groups using a representation-theoretic approach. By combining reduction theory, root- and weight-space analysis, and the mixing properties of diagonal flows, it derives precise Hausdorff-dimension formulas for the complement of ψ-Diophantine sets $S_ρ(\\psi)^c$, expressed in terms of the decay rate τ(\\psi), highest weights, and root data. The results yield both lower and upper bounds that culminate in sharp dimension formulas, including special cases for standard and adjoint representations, and recover known arithmetic cases. Connections to Diophantine approximation on flag varieties and Grassmannians are established, producing Jarník–Besicovitch-type theorems in these geometric settings and linking linear subspace approximation to representation-theoretic invariants. The work thus provides a unifying framework for shrinking target problems in higher-rank groups with broad implications for Diophantine geometry on algebraic varieties.

Abstract

In this paper, we study a shrinking target problem with target at infinity in a homogeneous space of a semisimple algebraic group from the representation-theoretic point of view. Let $ρ:\mathbf G\to\mathbf{GL}(V)$ be an irreducible $\mathbb Q$-rational representation of a connected semisimple $\mathbb Q$-algebraic group $\mathbf G$ on a complex vector space $V$, $\{a_t\}_{t\in\mathbb R}$ a one-parameter subgroup in a $\mathbb Q$-split torus in $\mathbf G$ and $ψ:\mathbb R_+\to\mathbb R_+$ a positive function on $\mathbb R_+$. We define a subset $S_ρ(ψ)$ of $ψ$-Diophantine elements in $\mathbf G(\mathbb R)$ in terms of the representation $ρ$ and $\{a_t\}_{t\in\mathbb R}$, and prove formulas for the Hausdorff dimension of the complement of $S_ρ(ψ)$. We also discuss the connections of our results to Diophantine approximation on flag varieties and rational approximation to linear subspaces in Grassmann varieties.

A shrinking target problem in homogeneous spaces of semisimple algebraic groups

TL;DR

This paper analyzes shrinking target problems at infinity on homogeneous spaces of connected semisimple algebraic groups using a representation-theoretic approach. By combining reduction theory, root- and weight-space analysis, and the mixing properties of diagonal flows, it derives precise Hausdorff-dimension formulas for the complement of ψ-Diophantine sets , expressed in terms of the decay rate τ(\\psi), highest weights, and root data. The results yield both lower and upper bounds that culminate in sharp dimension formulas, including special cases for standard and adjoint representations, and recover known arithmetic cases. Connections to Diophantine approximation on flag varieties and Grassmannians are established, producing Jarník–Besicovitch-type theorems in these geometric settings and linking linear subspace approximation to representation-theoretic invariants. The work thus provides a unifying framework for shrinking target problems in higher-rank groups with broad implications for Diophantine geometry on algebraic varieties.

Abstract

In this paper, we study a shrinking target problem with target at infinity in a homogeneous space of a semisimple algebraic group from the representation-theoretic point of view. Let be an irreducible -rational representation of a connected semisimple -algebraic group on a complex vector space , a one-parameter subgroup in a -split torus in and a positive function on . We define a subset of -Diophantine elements in in terms of the representation and , and prove formulas for the Hausdorff dimension of the complement of . We also discuss the connections of our results to Diophantine approximation on flag varieties and rational approximation to linear subspaces in Grassmann varieties.
Paper Structure (16 sections, 37 theorems, 418 equations)

This paper contains 16 sections, 37 theorems, 418 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction and main problem
  3. Main results
  4. Connections to Diophantine approximation
  5. Connection to Diophantine approximation on flag varieties
  6. Connection to rational approximation to linear subspaces
  7. Strategy of the proofs and organization of the paper
  8. Preliminaries
  9. Counting rational elements
  10. Proof of Theorem \ref{['mthm11']}: The case where $0\leq\tau(\psi)<\beta_0(a_{-1})$ and the flow $\{a_t\}_{t\in\mathbb R}$ is mixing
  11. Proof of Theorem \ref{['mthm11']}: The other cases
  12. An upper bound for the Hausdorff dimension of $S_\rho(\psi)^c$
  13. Proofs of Theorems \ref{['thm10']}, \ref{['thm11']} and \ref{['thm12']}
  14. Connections to Diophantine approximation
  15. Connection to Diophantine approximation on flag varieties
  16. ...and 1 more sections

Key Result

Theorem 1.3

We have $\tau(\psi)\geq0$ and $\beta_0(a_{-1})>0$. If $\tau(\psi)>\beta_0(a_{-1})$, then $S_\rho(\psi)^c=\emptyset$.

Theorems & Definitions (75)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 1.10: d21
  • ...and 65 more