For a real number x we define irrationality measure function as Ψ_x(t) = min_{1≤q≤t} ||qx||, where ||x|| is the distance to the nearest integer. Kan and Moshchevitin proved that for any x and y with x±y∉Z the difference of two irrationality measure functions Ψ_x(t)-Ψ_y(t) changes its sign infinitely many times as t↑∞. Later, Moshchevitin showed that there exists a constant C>0 such that |Ψ_x(t)-Ψ_y(t)|≥ C min{Ψ_x(t),Ψ_y(t)} for infinitely many t. It was also shown that C is optimal when both numbers are equal to the golden ratio up to a Möbius transformation with integer coefficients. We prove that this constant can be significantly improved when one of the numbers is not the golden ratio up to a Möbius transformation with integer coefficients. This is a joint work with Nikita Shulga from La Trobe University. We will also give a generalisation of Kan-Moshchevitin Theorem for $n > 2$ functions

# Irrationality measure functions

Abstract: