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Jarník-Besicovitch type theorems for semisimple algebraic groups

Published 14 Dec 2023 in math.DS and math.NT | (2312.09155v1)

Abstract: In this note, we initiate a study on Jarn\'ik-Besicovitch type theorems for semisimple algebraic groups from the representation-theoretic point of view. Let $\rho:\mathbf G\to\operatorname{\mathbf{GL}}(V)$ be an irreducible $\mathbb Q$-rational representation of a connected semisimple $\mathbb Q$-algebraic group $\mathbf G$ on a complex vector space $V$ and ${a_t}{t\in\mathbb R}$ a one-parameter subgroup in a $\mathbb Q$-split torus in $\mathbf G$. We define a subset $S\tau(\rho,{a_t}{t\in\mathbb R})$ of Diophantine elements of type $\tau$ in $\mathbf G(\mathbb R)$ in terms of the representation $\rho$ and the subgroup ${a_t}{t\in\mathbb R}$, and prove formulas for the Hausdorff dimension of the complement of $S_{\tau}(\rho,{a_t}_{t\in\mathbb R})$. As corollaries, we deduce several Jarn\'ik-Besicovitch type theorems.

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Authors (1)

  1. Cheng Zheng 

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