On Irregularity of Finite Sequences

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 $d$.