Sequences of Integers Dense in the Bohr Group

1 WHEN IS A SEQUENCE DENSE IN A COMPACT ABELIAN GROUP We denote by G a compact abelian group; we denote by μ its normalized Haar measure and by Γ = Ĝ its dual group (which is discrete). We consider sequences Λ ⊂ G and ask when is Λ dense in G. Our first approach is based on the duality theorem which tells us that the topology on G is the weak topology determined by Γ; thus if g0 ∈ G, then a basis for the neighborhoods of g0 in G is given by the sets ⋂k j=1{g : |〈g,γ j〉−〈g0,γ j〉|< ε} where k ∈ Z+, γ1 · · ·γk ∈ Γ and ε > 0. This proves Lemma 1.1. Λ is dense in G if, and only if, given k ∈ Z+, γ1 · · ·γk ∈ Γ, g0 ∈G, and ε > 0, there exists λ ∈ Λ such that