On irrationality measure functions for several real numbers

Abstract

. For n -tuple α = ( α 1 , . . . , α n ) of pairwise independent numbers we consider permutations of irrationality measure functions ψ α ( t ) = min 1 ≤ q ≤ t || qξ || . Let k ( α ) be the number of infinitely occurring different permutations { σ 1 , . . . , σ k ( α ) } . We prove that the size of n -tuple α with k ( α ) = k is n ≤ + . This result is optimal.