Exact Generic Exponents of Approximation for the Action of the Integral Special Linear Group on the Torus
Abstract
Given an integer n≥ 2 and points x, y lying in the n--dimensional torus 𝕋n equipped with its usual metric denoted by ⟨ · ⟩, let τn(x,y) = sup{ τ≥0: l there exist infinitely many H≥1 for which [1mm] |A|≤ H and ⟨ Ax−y⟩≤ H−τ for some A∈SLn(ℤ) } be the Diophantine exponent of asymptotic approximation of y by the orbit of x under the action of the integral special linear group SLn(ℤ). Let also τ̂n (x, y) = sup{ τ≥0: l for all but finitely many H≥1, there is A∈SLn(ℤ) [1mm] such that |A|≤ H and ⟨ Ax−y⟩≤ H−τ } denote the corresponding exponent of uniform approximation. The main result of this paper proves that for almost all x, y∈𝕋n (with respect to the induced Lebesgue measure), τ̂n (x, y) = τn (x, y) = n−1. This is the first result to establish the exact generic approximation exponents for an action of a lattice subgroup of SLn(ℝ) induced by the standard Euclidean action.
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BibTeX
@misc{adiceam2026exact,
title = {{Exact Generic Exponents of Approximation for the Action of the Integral Special Linear Group on the Torus}},
author = {Faustin Adiceam and Régis Aolo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36397},
eprint = {2610.36397},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36397},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Exact Generic Exponents of Approximation for the Action of the Integral Special Linear Group on the Torus
AU - Faustin Adiceam
AU - Régis Aolo
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36397
UR - https://arxiv.org/abs/2610.36397
AB - Given an integer n≥ 2 and points x, y lying in the n--dimensional torus 𝕋^n equipped with its usual metric denoted by ⟨ · ⟩, let τ_n(x,y) = sup{ τ≥0: l there exist infinitely many H≥1 for which [1mm] |A|≤ H and ⟨ Ax−y⟩≤ H^−τ for some A∈SL_n(ℤ) } be the Diophantine exponent of asymptotic approximation of y by the orbit of x under the action of the integral special linear group SL_n(ℤ). Let also τ̂_n (x, y) = sup{ τ≥0: l for all but finitely many H≥1, there is A∈SL_n(ℤ) [1mm] such that |A|≤ H and ⟨ Ax−y⟩≤ H^−τ } denote the corresponding exponent of uniform approximation. The main result of this paper proves that for almost all x, y∈𝕋^n (with respect to the induced Lebesgue measure), τ̂_n (x, y) = τ_n (x, y) = n−1. This is the first result to establish the exact generic approximation exponents for an action of a lattice subgroup of SL_n(ℝ) induced by the standard Euclidean action.
ER -