On \(L_ \infty\) universally bad sequences in ergodic theory (Q1358736)

The author proves an arithmetical condition (conditions \((c)\) or \((d)\) of the Theorem) for a subsequence of the positive integers to be \(L_\infty\) universally bad. As a consequence, he proves (Corollary 1) that every \(L_\infty\) universally bad sequence is \(\delta\)-sweeping out for some \(\delta>0\). This problem was posed by \textit{J. Rosenblatt} [Almost everywhere convergence II, Proc. 2nd Int. Conf., Evanston/IL (USA) 1989, 227-245 (1991; Zbl 0745.28011), p. 231] and \textit{A. Bellow} and \textit{R. L. Jones} [Adv. Math. 120, No. 1, 155-172 (1996; Zbl 0878.46020)] also solve it by a different method. Corollary 2 shows that one can test \(L_\infty\) universal badness of sequences on the special dynamical system \(([0,1],B,\lambda,x\to 2x\pmod 1)\) consisting of Borel sets of \([0,1]\) with the Lebesgue measure, and transformation \(x\to 2x\pmod 1\).