Geodesic
Local Discrepancies in the Problem of the Distribution of the Sequence~$\{k\alpha\}$
Matematičeskie zametki, Tome 109 (2021) no. 3, pp. 452-463

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The paper deals with local discrepancies in the problem of the distribution of the sequence $\{k\alpha\}$, i.e., the remainder terms in asymptotic formulas for the number of points in this sequence lying in prescribed intervals. The construction of intervals for which local discrepancies tend to infinity slower than any given function is presented. Moreover, it is shown that there exists an uncountable set of such intervals. Previously, similar results were obtained only for irrationalities with bounded partial quotients of their continued fraction expansions.
Keywords: uniform distribution, local discrepancies, inhomogeneous Diophantine approximations, continued fractions. 
@article{MZM_2021_109_3_a10,
     author = {A. V. Shutov},
     title = {Local {Discrepancies} in the {Problem} of the {Distribution} of the {Sequence~}$\{k\alpha\}$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {452--463},
     publisher = {mathdoc},
     volume = {109},
     number = {3},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2021_109_3_a10/}
}
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A. V. Shutov. Local Discrepancies in the Problem of the Distribution of the Sequence~$\{k\alpha\}$. Matematičeskie zametki, Tome 109 (2021) no. 3, pp. 452-463. http://geodesic.mathdoc.fr/item/MZM_2021_109_3_a10/