On two-dimensional Dirichlet spectrum

2013/06/08 by Renat Akhunzhanov, Renat K. Akhunzhanov, Akhunzhanov, Renat +3
Mathematics · #11J06 #FOS: Mathematics #Number Theory (math.NT) #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #advanced mathematical theories #math.NT #msc:11J06

paper · pdf · doi:10.48550/arxiv.1306.1876

In Russian, summary in English. Sbmitted to Moscow journal of Combinatorisc and Number Theory

arxiv created 2013/06/08 · openalex publication_date 2013/06/08 · arxiv updated 2013/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define two-dimensional Dirichlet spectrum (with respect to Euclidean norm) as D₂=λ\inR | ∃ v=(v₁,v₂)∈ \mathbf R²: \limsup_t→∞ t⋅ψ_v²(t)=λ, where ψ_v(t)=min_{1\leqslant q\leqslant t}√(|q v₁|²+|q v₂|²) is the two-dimensional "irrationality measure function". Our main result states the equality D₂=[0; 2/sqrt3].

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