On an inequality of Erdős and Turán concerning uniform distribution modulo one. II (Q1336059)

[Part I, cf. Colloq. Math. Soc. Janos Bolyai 60, 621-630 (1992; Zbl 0791.60013).] Let \(x_ 1, \dots, x_ N\) be a sequence of real numbers; \(\Delta_ N\) its discrepancy and \(\alpha_ k = {1 \over N} \sum^ N_{j=1} e^{2 \pi ik}\) its Fourier coefficients. A famous theorem of Erdős and Turán says \[ \Delta_ N \ll B_ N : = \min_ K \left( {1 \over k} + \sum^{K-1}_{k=1} {| \alpha_ k | \over k} \right). \] The author proves the converse inequality \(\Delta_ N \gg B_ N^{3/2}\) and shows that this bound is best possible by constructing a point set satisfying \(\Delta_ N \ll B_ N^{3/2}\). The construction is based on the Rudin-Shapiro polynomials.