Oscillation of irrational measure function in the multidimensional case

TL;DR

This paper investigates the oscillatory behavior of the difference of irrational measure functions in multidimensional cases, proving infinite sign changes for almost all pairs of parameters in specific dimensions.

Contribution

It establishes that the difference function between irrational measure functions oscillates infinitely often in certain multidimensional settings for almost all parameter pairs.

Findings

Difference function changes sign infinitely often as t approaches infinity.

Oscillation occurs for almost all pairs of parameters in specified dimensions.

Results extend understanding of irrational measure function behavior in multidimensional cases.

Abstract

We proved that difference function $\psi_\Theta-\psi_{\Theta'}$ for almost all pairs $\Theta$, $\Theta'$ in cases $m=1$, $n=2$ or $m\geqslant2$ and $n=1$ changes its sign infinity many times as $t\rightarrow+\infty$.