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arXiv · 1306.1876

On two-dimensional Dirichlet spectrum

Renat Akhunzhanov·Denis Shatskov
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Abstract

We define two-dimensional Dirichlet spectrum (with respect to Euclidean norm) as D_2=\lambda\in\mathbf{R} | \exists \mathbf{v}=(v_1,v_2)\in \mathbf {R}^2: \limsup\limits_{t\rightarrow\infty} {t\cdot\psi_{v}^2(t)}=\lambda, where \psi_{v}(t)=\min\limits_{1\leqslant q\leqslant t}\sqrt{|q v_1|^2+|q v_2|^2} is the two-dimensional "irrationality measure function". Our main result states the equality D_2=[0; 2/sqrt{3}].

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Renat Akhunzhanov, Denis Shatskov. 2013-06-08. On two-dimensional Dirichlet spectrum. https://arxiv.org/abs/1306.1876

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