On irregularity of finite sequences (Q2234364)

Let \(s(d)\) be the maximum \(N\) for which there exists a sequence \((x_1,x_2,\ldots,x_{N+d})\) of numbers in \([0;1)\) such that for every \(n=1,2,\ldots,N\) each of the intervals \([0,1), [1,2), \ldots, [n-1,n)\) contains at least on element of the sequence \((nx_1,nx_2,\ldots,nx_{n+d})\). Then it is proved that \(s(d)\geq 2d\). Corresponding \(x_i\) are \(x_i=\frac{i-1}{d}\) for \(1\le i\le d\), \(x_i=\frac{2i-2d-1}{2d}\) for \(d+1\le i\le 2d\), also we can take arbitrary \(x_i\) for \(2d+1\le i\le 3d\). Also it is proved that for \(d\ge 4\cdot 10^{16}\) \(s(d) <200d\). After the publication of this paper Anholcer, Bosek, Grytczuk, Gutowski, Przybyło, Pyzik and Zając announce that \(\lfloor \frac{\ln 2}{1-\ln 2}d\rfloor \leq s(d)\leq \frac{1+\ln 2}{1-\ln 2}d+o(1)\) [\textit{M. Anholcer} et al., ``On a problem of Steinhaus, Preprint, \url{arXiv:2111.01887}].