On sequences with prescribed metric discrepancy behavior (Q331057)

By a result of \textit{R. C. Baker} [J. Lond. Math. Soc., II. Ser. 24, 34--40 (1981; Zbl 0422.10048)], the discrepancy \(D_{N}=D_{N}(a_{n}\alpha)\) of the sequence \((a_{n}\alpha)_{n\geq1}\) for a given sequence of distinct positive integers \((a_{n})\) and almost all \(\alpha\) satisfies the bound \[ N D_{N}=\mathcal{O}(N^{\frac{1}{2}}(\log N)^{\frac{3}{2}+\varepsilon}), \] for all \(\varepsilon>0\). The authors of the present paper show that in the range \((0,\frac{1}{2}]\) any convergence rate for the metric behaviour of the discrepancy can be attained. More precisely, they prove the following result. Let \(0<\gamma\leq\frac{1}{2}\). Then there exists a strictly increasing sequence \((a_{n})_{n\geq 1}\) of positive integers such that for the discrepancy of the sequence \((\{a_{n}\alpha\})_{n\geq 1}\) for almost all \(\alpha\) we have \[ N D_{N}=\mathcal{O}(N^{\gamma}) \] and \[ N D_{N}=\Omega(N^{\gamma-\varepsilon}) \] for all \(\varepsilon>0\).