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Fourier dimension for exactly approximable matrices
Abstract
We determine the Fourier dimension of exactly approximation set for all dimensions. More precisely, let m,n≥1, θ∈ℝm, and ψ:ℕ→ ℝ+ be a non-increasing approximate function. Denote by Exactm,n(ψ,θ) the set of m× n matrices which are inhomogeneously ψ-approximable with shift θ, but not cψ-approximable for any fixed 0<c<1. We prove that dimF Exactm,n(ψ,θ)=2n/(1+τψ), whenever τψ>n/m, where τψ is the lower order of ψ at infinity.
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BibTeX
@misc{he2026fourier,
title = {{Fourier dimension for exactly approximable matrices}},
author = {Yubin He and Bixuan Li and Baowei Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23680},
eprint = {2610.23680},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23680},
}
RIS
TY - UNPB
TI - Fourier dimension for exactly approximable matrices
AU - Yubin He
AU - Bixuan Li
AU - Baowei Wang
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23680
UR - https://arxiv.org/abs/2610.23680
AB - We determine the Fourier dimension of exactly approximation set for all dimensions. More precisely, let m,n≥1, θ∈ℝ^m, and ψ:ℕ→ ℝ^+ be a non-increasing approximate function. Denote by Exact_m,n(ψ,θ) the set of m× n matrices which are inhomogeneously ψ-approximable with shift θ, but not cψ-approximable for any fixed 0<c<1. We prove that dim_F Exact_m,n(ψ,θ)=2n/(1+τ_ψ), whenever τ_ψ>n/m, where τ_ψ is the lower order of ψ at infinity.
ER -