Harmonious set
In mathematics, a harmonious set is a subset of a locally compact abelian group on which every weak character may be uniformly approximated by strong characters. Equivalently, a suitably defined dual set is relatively dense in the Pontryagin dual of the group. This notion was introduced by Yves Meyer in 1970 and later turned out to play an important role in the mathematical theory of quasicrystals. Some related concepts are model sets, Meyer sets, and cut-and-project sets. Definition. Let "Λ" be a subset of a locally compact abelian group "G" and "Λ" be the subgroup of  G generated by "Λ", with discrete topology. A weak character is a restriction to "Λ" of an algebraic homomorphism from "Λ" into the circle group: A strong character is a restriction to "Λ" of a continuous homomorphism from "G" to T, that is an element of the Pontryagin dual of "G". A set "Λ" is harmonious if every weak character may be approximated by strong characters uniformly on "Λ". Thus for any "ε" > 0 and any weak character "χ", there exists a strong character "ξ" such that If the locally compact abelian group "G" is separable and metrizable (its  topology may be defined by a translation-invariant metric) then harmonious sets admit another, related, description. Given a subset "Λ" of "G" and a positive "ε", let "M" be the subset of the Pontryagin dual of "G" consisting of all characters that are almost trivial on "Λ": Then "Λ" is harmonious if the sets "M" are relatively dense in the sense of Besicovitch: for every "ε" > 0 there exists a compact subset "K" of the Pontryagin dual such that Properties. The next two properties show that the notion of a harmonious set is nontrivial only when the ambient group  is neither compact nor discrete. Examples. Interesting examples of multiplicatively closed harmonious sets of real numbers arise in the theory of diophantine approximation.