Vol. 11, No. 1, 2022
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Rational approximations to two irrational numbers

Nikita Shulga
Vol. 11 (2022), No. 1, 1–10
DOI: 10.2140/moscow.2022.11.1
Abstract

For real ξ we consider the irrationality measure function ψξ(t) = min ⁡ 1≤q≤t,q∈ℤ∥qξ∥. We prove that in the case α ± β∉ℤ there exist arbitrary large values of t with

| 1 ψα(t) − 1 ψβ(t)| ≥5( 1 −5 − 1 2 )t.

The constant on the right-hand side is optimal.

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Keywords
irrationality measure function, diophantine approximation, continued fractions
Mathematical Subject Classification
Primary: 11J06, 11J70
Milestones
Received: 7 April 2021
Revised: 9 June 2021
Accepted: 25 June 2021
Published: 30 March 2022
Authors
Nikita Shulga
Moscow Center for Fundamental and Applied Mathematics
Moscow
Russia