Lagrange spectrum for Diophantine approximations of complex numbers with real part equal to one half
Abstract
Motivated by a theorem of A. Schmidt on the part of the complex Lagrange spectrum below $2$, we study the restricted Lagrange spectrum L1/2+iℝ arising from the approximation of complex numbers of the form 1/2+iα, α∈ℝ∖ℚ by Gaussian rationals p/q with p,q∈ℤ[i], q≠ 0. We show that this spectrum admits a description in terms of a dynamical spectrum associated with a real horseshoe. As a consequence, we obtain several fractal properties of L1/2+iℝ, such as continuity of the dimension function t↦dimH(L1/2+iℝ∩(−∞,t)). We also prove that the set of complex numbers z satisfying lvert z−p/qrvert≥1/2|q|2, for all p,q∈ℤ[i], q≠ 0, is uncountable. In fact, we show that this inequality holds for every complex number of the form z=1/2(1+iθ) where θ∈ℝ∖ℚ is a root of one of Schmidt's C-minimal forms.
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BibTeX
@misc{cheng2026lagrange,
title = {{Lagrange spectrum for Diophantine approximations of complex numbers with real part equal to one half}},
author = {Hao Cheng and Harold Erazo and Carlos Gustavo Moreira and Thiago Vasconcelos},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29527},
eprint = {2608.29527},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29527},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Lagrange spectrum for Diophantine approximations of complex numbers with real part equal to one half
AU - Hao Cheng
AU - Harold Erazo
AU - Carlos Gustavo Moreira
AU - Thiago Vasconcelos
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29527
UR - https://arxiv.org/abs/2608.29527
AB - Motivated by a theorem of A. Schmidt on the part of the complex Lagrange spectrum below $2$, we study the restricted Lagrange spectrum L_1/2+iℝ arising from the approximation of complex numbers of the form 1/2+iα, α∈ℝ∖ℚ by Gaussian rationals p/q with p,q∈ℤ[i], q≠ 0. We show that this spectrum admits a description in terms of a dynamical spectrum associated with a real horseshoe. As a consequence, we obtain several fractal properties of L_1/2+iℝ, such as continuity of the dimension function t↦dim_H(L_1/2+iℝ∩(−∞,t)). We also prove that the set of complex numbers z satisfying lvert z−p/qrvert≥1/2|q|^2, for all p,q∈ℤ[i], q≠ 0, is uncountable. In fact, we show that this inequality holds for every complex number of the form z=1/2(1+iθ) where θ∈ℝ∖ℚ is a root of one of Schmidt's C-minimal forms.
ER -