Geodesic
Oscillation of the Measure of Irrationality Function in the Multidimensional Case
Matematičeskie zametki, Tome 99 (2016) no. 1, pp. 102-120

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It is proved that, for almost all pairs of $n\times m$ matrices $\Theta$, $\Theta'$, in the cases $m=1$ and $n=2$ or $m\ge2$ and $n=1$, the difference between the measure of irrationality functions $\psi_\Theta-\psi_{\Theta'}$ oscillates an infinite number of times as $t\to+\infty$.
Keywords: measure of irrationality function of a matrix, oscillation of a function, algebraically independent real numbers, Borel–Cantelli sequence.
Mots-clés : Lebesgue measure 
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     author = {D. O. Shatskov},
     title = {Oscillation of the {Measure} of {Irrationality} {Function} in the {Multidimensional} {Case}},
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D. O. Shatskov. Oscillation of the Measure of Irrationality Function in the Multidimensional Case. Matematičeskie zametki, Tome 99 (2016) no. 1, pp. 102-120. http://geodesic.mathdoc.fr/item/MZM_2016_99_1_a9/