Further Baire Results on the Distribution of Subsequences

This paper presents results about the distribution of subsequences which are typical in the sense of Baire. The first main part is concerned with sequences of the type xk = nkα, n1 < n2 < n3 < · · · , mod 1. Improving a result of Šalát we show that, if the quotients qk = nk+1/nk satisfy qk ≥ 1 + ε, then the set of α such that (xk) is uniformly distributed is of first Baire category, i.e. for generic α we do not have uniform distribution. Under the stronger assumption limk→∞ qk = ∞ one even has maldistribution for generic α, the strongest possible contrast to uniform distribution. Nevertheless, growth conditions on the nk alone do not suffice to explain various interesting phenomena. In particular, for individual sequences the situation maybe quite diverse: For nk = 2 k there is a set M such that for generic α the set of all limit measures of (xk) is exactly M , while for nk = 2 k + 1 such an M does not exist. For the rest of the paper we consider 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) ∈ S the set of limit measures of the subsequence (xnk ) is exactly M .