We prove an arithmetical condition (conditions (c) or (d) of the theorem) for a subsequence of the positive integers to be L_∞ universally bad. As a consequence, we prove (Corollary 1) that every L_∞ universally bad sequence is δ-sweeping out for some δ > 0. This problem was posed by J. Rosenblatt [2, p. 231] and (as I was informed by R. Jones and J. Rosenblatt) A. Bellow and R. Jones also solve it by a different method in [1], as a corollary of their main result there. Our Corollary 2 shows that one can test L_∞ universal badness of sequences on the special dynamical system ([0, 1], B, λ, x → 2x(mod 1)) consisting of Borel sets of [0, 1] with the Lebesgue measure, and transformation x → 2x(mod 1).
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