Geodesic
On the Mean Value of the Measure of Irrationality of Real Numbers
Matematičeskie zametki, Tome 98 (2015) no. 2, pp. 271-287

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This paper deals with the asymptotic behavior of the integral $$ I_\alpha(t)=\int_1^t \psi_\alpha(\xi)\,d\xi, \qquad\text{where}\quad \psi_\alpha(t)=\min_{1\le q\le t}\|q\alpha\| $$ (here the minimum is taken over integers $q$ and $\|\,\cdot\,\|$ denotes the distance to the nearest integer).
Keywords: real number, measure of irrationality, continued fraction
Mots-clés : convergent, Lebesgue measure, Gauss transformation, ergodic transformation. 
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     author = {D. O. Shatskov},
     title = {On the {Mean} {Value} of the {Measure} of {Irrationality} of {Real} {Numbers}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {271--287},
     publisher = {mathdoc},
     volume = {98},
     number = {2},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2015_98_2_a11/}
}
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D. O. Shatskov. On the Mean Value of the Measure of Irrationality of Real Numbers. Matematičeskie zametki, Tome 98 (2015) no. 2, pp. 271-287. http://geodesic.mathdoc.fr/item/MZM_2015_98_2_a11/