Further Baire results on the distribution of subsequences
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This paper presents results about the distribution of subsequences which are typical in the sense of Baire.
The first part is concerned with sequences of the type x_k = n_k*alpha, n_1 < n_2 < n_3 < ..., mod 1. Improving a result of Salat we show that, if the quotients q_k = n_{k+1}/n_k satisfy q_k > 1+ epsilon, then the set of alpha such that (x_k) is uniformly distributed is of first Baire category, i.e. for generic alpha we do not have uniform distribution. Under the stronger assumption lim q_k = infinity one even has maldistribution for generic alpha, the strongest possible contrast to uniform distribution.
The second part reverses the point of view by considering appropriately defined Baire spaces S of subsequences. For a fixed well distributed sequence (x_n) we show that there is a set M of measures such that for generic (n_k) in S the set of limit measures of the subsequence (x_{n_k}) is exactly M.
21 pages, LaTeX2e. Final version. (Somewhat expanded proofs and clarifications, more examples)
21 pages, LaTeX2e. Final version. (Somewhat expanded proofs and clarifications, more examples)