Harmonious set

Let Λ be a subset of a locally compact abelian group G and Λ_d 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_d into the circle group:

${\displaystyle \chi$ :\Lambda _{d}\to \mathbf {T} ,\quad \chi \in \operatorname {Hom} (\Lambda _{d},\mathbf {T} ).}

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

${\displaystyle \sup _{\Lambda }|\chi (\lambda )-\xi (\lambda )|\leq \epsilon ,\quad \chi \in \operatorname {Hom} (\Lambda _{d},\mathbf {T} ),\xi \in {\hat {G}}.}$

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_ε be the subset of the Pontryagin dual of G consisting of all characters that are almost trivial on Λ:

${\displaystyle \sup _{\Lambda }|\chi (\lambda )-1|\leq \epsilon ,\quad \chi \in {\hat {G}}.}$

Then Λ is harmonious if the sets M_ε are relatively dense in the sense of Besicovitch: for every ε > 0 there exists a compact subset K_ε of the Pontryagin dual such that

${\displaystyle M_{\epsilon }+K_{\epsilon }={\hat {G}}.}$

• A subset of a harmonious set is harmonious.
• If Λ is a harmonious set and F is a finite set then the set Λ + F is also harmonious.

The next two properties show that the notion of a harmonious set is nontrivial only when the ambient group is neither compact nor discrete.

• A finite set Λ is always harmonious. If the group G is compact then, conversely, every harmonious set is finite.
• If G is a discrete group then every set is harmonious.

Interesting examples of multiplicatively closed harmonious sets of real numbers arise in the theory of diophantine approximation.

• Let G be the additive group of real numbers, θ >1, and the set Λ consist of all finite sums of different powers of θ. Then Λ is harmonious if and only if θ is a Pisot number. In particular, the sequence of powers of a Pisot number is harmonious.
• Let K be a real algebraic number field of degree n over Q and the set Λ consist of all Pisot or Salem numbers of degree n in K. Then Λ is contained in the open interval (1,∞), closed under multiplication, and harmonious. Conversely, any set of real numbers with these 3 properties consists of all Pisot or Salem numbers of degree n in some real algebraic number field K of degree n.

• Almost periodic function

• Yves Meyer, Algebraic numbers and harmonic analysis, North-Holland Mathematical Library, vol.2, North-Holland, 1972