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| 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 [[quasicrystal]]s. Some related concepts are '''model sets''', '''[[Meyer set]]s''', and '''cut-and-project sets'''.
| | I'm Rachelle and I live in Foxground. <br>I'm interested in Creative Writing, Nordic skating and Italian art. I like to travel and watching Bones.<br><br>my blog post - [http://www.bataviarr.com/profile/sahoule Biking for modern life mountain bike sizing.] |
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| == Definition ==
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| Let ''Λ'' be a subset of a locally compact abelian group ''G'' and ''Λ''<sub>''d''</sub> 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''<sub>''d''</sub> into the [[circle group]]:
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| : <math> \chi: \Lambda_d\to\mathbf{T}, \quad
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| \chi\in\operatorname{Hom}(\Lambda_d,\mathbf{T}). </math>
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| 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''.
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| A set ''Λ'' is '''harmonious''' if every weak character may be approximated by
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| strong characters uniformly on ''Λ''. Thus for any ''ε'' > 0 and any weak character ''χ'', there exists a strong character ''ξ'' such that
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| : <math> \sup_\Lambda |\chi(\lambda)-\xi(\lambda)| \leq \epsilon, \quad
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| \chi\in\operatorname{Hom}(\Lambda_d,\mathbf{T}), \xi\in\hat{G}. </math>
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| If the locally compact abelian group ''G'' is [[separable topological space|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''<sub>''ε''</sub> be the subset of the Pontryagin dual of ''G'' consisting of all characters that are almost trivial on ''Λ'':
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| : <math> \sup_\Lambda|\chi(\lambda)-1| \leq \epsilon, \quad
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| \chi\in\hat{G}.</math>
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| Then ''Λ'' is '''harmonious''' if the sets ''M''<sub>''ε''</sub> are '''relatively dense''' in the sense of [[Besicovitch]]: for every ''ε'' > 0 there exists a compact subset ''K''<sub>''ε''</sub> of the Pontryagin dual such that
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| : <math> M_\epsilon + K_\epsilon = \hat{G}.</math>
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| == Properties ==
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| * A subset of a harmonious set is harmonious.
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| * If ''Λ'' is a harmonious set and ''F'' is a finite set then the set ''Λ'' + ''F'' is also harmonious.
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| The next two properties show that the notion of a harmonious set is nontrivial only when the ambient group is neither compact nor discrete.
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| * A finite set ''Λ'' is always harmonious. If the group ''G'' is compact then, conversely, every harmonious set is finite.
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| * If ''G'' is a [[discrete group]] then every set is harmonious.
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| == Examples ==
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| Interesting examples of multiplicatively closed harmonious sets of real numbers arise in the theory of [[diophantine approximation]].
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| * Let ''G'' be the additive group of [[real number]]s, ''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.
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| * Let '''K''' be a real [[algebraic number field]] of degree ''n'' over '''Q''' and the set ''Λ'' consist of all Pisot or [[Salem number|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''.
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| == See also ==
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| * [[Almost periodic function]]
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| == References ==
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| * [[Yves Meyer]], ''Algebraic numbers and harmonic analysis'', North-Holland Mathematical Library, vol.2, North-Holland, 1972
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| [[Category:Harmonic analysis]]
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| [[Category:Diophantine approximation]]
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| [[Category:Tessellation]]
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I'm Rachelle and I live in Foxground.
I'm interested in Creative Writing, Nordic skating and Italian art. I like to travel and watching Bones.
my blog post - Biking for modern life mountain bike sizing.