Geodesic
On Irregularity of Finite Sequences
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and Combinatorial Number Theory, Tome 314 (2021), pp. 97-102.

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A sequence $(x_1,x_2,\dots ,x_{N+d})$ of numbers in $[0,1)$ is said to be $N$-regular with at most $d$ irregularities if for every $n=1,\dots ,N$ each of the intervals $[0,1),[1,2),\dots ,[n-1,n)$ contains at least one element of the sequence $(nx_1,nx_2,\dots ,nx_{n+d})$. The maximum $N$ for which there exists an $N$-regular sequence with at most $d$ irregularities is denoted by $s(d)$. We show that $s(d)\ge 2d$ for any positive integer $d$ and that $s(d)200d$ for all sufficiently large $d$.
Keywords: distribution of sequences of real numbers. 
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     title = {On {Irregularity} of {Finite} {Sequences}},
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S. V. Konyagin. On Irregularity of Finite Sequences. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and Combinatorial Number Theory, Tome 314 (2021), pp. 97-102. http://geodesic.mathdoc.fr/item/TM_2021_314_a4/

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