An improved characterization of normal sets and some counter-examples

We strengthen G. Rauzy’s characterization of normal sets by observing that the so-called elementary sets are precisely the F_σδ sets. This answers in the affirmative Rauzy’s open question: Are finite unions of normal sets necessarily normal? We also generalize the notion and characterization of normal sets from subsets of ? to subsets of ? ^d . This allows us to answer a question of E. Lesigne and M. Wierdl with the following construction: There exist two sequences of real numbersU=(u _n ) _nε?,V=(v _n ) _nε? such thatαU+βV=(αu _n +βv _n ) _nε? is uniformly distributed mod 1 if and only if exactly one of the real numbers α, β vanishes. Additionally, we provide the ‘ultimate’ counter-example (stronger than that of H. G. Meijer and R. Sattler) to a conjecture of S. Uchiyama by constructing a sequence of integersU which is u.d. in ? (i.e. u.d. modk for allk ε ?), but such that for all realα, αU is not u.d. mod 1.