Difference of irrationality measure functions

2023/05/17 by Viktoria Rudykh, Rudykh, Viktoria, Nikita Shulga +1 · 2 citations
Mathematics · #11A55 #11J13 #Analytic Number Theory Research #FOS: Mathematics #Functional Equations Stability Results #History and Theory of Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2305.10264

openalex publication_date 2023/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For an irrational number α∈ℝ we consider its irrationality measure function ψ_α(x) = min_{1≤ q≤ x, q∈ℤ} ‖ qα‖. It is known for all irrational numbers α and β satisfying α±β\not∈ℤ, there exist arbitrary large values of t with | ψ_α(t) - ψ_β(t) | \geqslant ( √τ - 1) ⋅ min( ψ_α(t), ψ_β(t) ), where τ= (√(5) + 1)/(2) and this result is optimal for certain numbers equivalent to τ. Here we prove that for all irrational numbers α and β, satisfying α±β\not∈ℤ, such that at least one of them is not equivalent to τ, there exist arbitrary large values of t with | ψ_α(t) - ψ_β(t) | \geqslant (√(√2+1)-1)⋅ min( ψ_α(t), ψ_β(t) ). Moreover, we show that the constant on the right-hand side is optimal.

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