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On the Growth of Denominators of Simultaneous Best Diophantine Approximations in a Norm Induced by an Inner Product
Abstract
For n-dimensional simultaneous best Diophantine approximations in an arbitrary norm induced by an inner product, for n≥2 we prove qk+2n≥ qk+min{qk+2n−1,2qk+1}. This yields gn(α):=liminfm→∞(qm)1/m≥φ1/2n−1, for φ= (1 + √5)/2. Consequently, G(n):=infα∈ℝn∖ℚngn(α)≥φ1/2n−1 and underline𝒟n(α)≤⌊ 2n−1log2/logφ⌋+1, where underline𝒟n(α) is a quantity related to multidimensional generalizations of the three-distance theorem. In particular, g2(α)≥√φ, g3(α)≥√[4]φ, underline𝒟2(α)≤3, underline𝒟3(α)≤6.
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BibTeX
@misc{shatunov2026growth,
title = {{On the Growth of Denominators of Simultaneous Best Diophantine Approximations in a Norm Induced by an Inner Product}},
author = {Leonid M. Shatunov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02386},
eprint = {2609.02386},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02386},
}
RIS
TY - UNPB
TI - On the Growth of Denominators of Simultaneous Best Diophantine Approximations in a Norm Induced by an Inner Product
AU - Leonid M. Shatunov
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.02386
UR - https://arxiv.org/abs/2609.02386
AB - For n-dimensional simultaneous best Diophantine approximations in an arbitrary norm induced by an inner product, for n≥2 we prove q_k+2^n≥ q_k+min{q_k+2^n−1,2q_k+1}. This yields g_n(α):=liminf_m→∞(q_m)^1/m≥φ^1/2^n−1, for φ= (1 + √5)/2. Consequently, G(n):=inf_α∈ℝ^n∖ℚ^ng_n(α)≥φ^1/2^n−1 and underline𝒟_n(α)≤⌊ 2^n−1log2/logφ⌋+1, where underline𝒟_n(α) is a quantity related to multidimensional generalizations of the three-distance theorem. In particular, g_2(α)≥√φ, g_3(α)≥√[4]φ, underline𝒟_2(α)≤3, underline𝒟_3(α)≤6.
ER -