Further Baire results on the distribution of subsequences
Abstract
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 xk = nk*alpha, n1 < n2 < n3 < ..., mod 1. Improving a result of Salat we show that, if the quotients qk = nk+1/nk satisfy qk > 1+ epsilon, then the set of alpha such that (xk) is uniformly distributed is of first Baire category, i.e. for generic alpha we do not have uniform distribution. Under the stronger assumption lim qk = 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 (xn) we show that there is a set M of measures such that for generic (nk) in S the set of limit measures of the subsequence (xnk) is exactly M.
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