A shrinking target problem in homogeneous spaces of semisimple algebraic groups
Year: 2026
Type: article
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 : 𝐆 → 𝐆 𝐋 ( 𝑉 ) be an irreducible ℚ -rational representation of a connected semisimple ℚ -algebraic group G on a complex vector space V , { 𝑎 𝑡 } 𝑡 ∈ ℝ a one-parameter subgroup in a ℚ -split torus in G and 𝜓 : ℝ + → ℝ + a positive function on ℝ + . We define a subset 𝑆 𝜌 ( 𝜓 ) of ψ -Diophantine elements in 𝐆 ( ℝ ) in terms of the representation ρ and { 𝑎 𝑡 } 𝑡 ∈ 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.
Source: Advances in Mathematics
Author: Cheng Zheng
Institution: Shanghai Jiao Tong University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,790
OpenAlex ID: W4389821258