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 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.