Vol. 11, No. 2, 2022
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On irrationality measure functions for several real numbers

Viktoria Rudykh
Vol. 11 (2022), No. 2, 197–204
DOI: 10.2140/moscow.2022.11.197
Abstract

For an n-tuple α = (α1,… ⁡,αn) of pairwise independent numbers we consider permutations

σ(t) : {1,2,3,… ⁡,n}→{σ1,σ2,σ3,… ⁡,σn}, ψασ 1(t) > ψασ 2(t) > ψασ 3(t) > ⋯ > ψασn(t),

of irrationality measure functions ψα(t) = min ⁡ 1≤q≤t∥qα∥. Let 𝔨(α) be the number of infinitely occurring different permutations {σ1,… ⁡,σ𝔨(α)}. We prove that the length of an n-tuple α with 𝔨(α) = k is

n ≤ k(k + 1) 2

and this result is optimal.

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Keywords
irrationality measure function
Mathematical Subject Classification
Primary: 11J13
Milestones
Received: 27 April 2022
Revised: 24 June 2022
Accepted: 8 July 2022
Published: 13 August 2022
Authors
Viktoria Rudykh
Moscow Institute of Physics and Technology
Dolgoprudny
Russia