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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'''.


== Definition ==


Let ''&Lambda;'' be a subset of a locally compact abelian group ''G'' and ''&Lambda;''<sub>''d''</sub> be the subgroup of ''G'' generated by ''&Lambda;'', with [[discrete topology]]. A '''weak character''' is a restriction to ''&Lambda;'' of an algebraic homomorphism from ''&Lambda;''<sub>''d''</sub> into the [[circle group]]:
Your assignment is to plan your work in the right way, sticking with our guidelines:. Use them at the beginning and end of your paragraphs. In writing College Application Essay, the students may also tell about themselves that will capture the attention of the readers. We strive to be a world essay writing company through providing quality services at a price which is equivalent with our services. Now the writing an essay has become the big issue because if the person is not writing an essay of his or her own or for any body then this does not mean only that he or she do not know how to write it down but the thing is that there are many circumstances because of which most of the people cannot write it down. <br><br>Definition essay- A definition essay defines a particular term. We have brilliant experts who have the required skills in essay writing. Paragraphs 2, 3, and 4: each paragraph covers one of the three points outlined in paragraph 1. Students are also advised to write about what they find passionate in their lives. Custom essay writing may cause the sceptic in you to raise the alarm. <br><br>They can ask themselves lots of questions and write down the answers. My passion for food began more than twenty years ago when I was a secondary school student. As multiple readers generates multiple thoughts adding variety to your perception. If you order essays for sale by them, you will definitely be completely satisfied. Article Source: you have difficulties with writing essays, pay a visit to essaysprofy. <br><br>Cornell is very clear there about what is required. When you loved this informative article and you would want to receive more information regarding [http://tapety24.biz.pl/profile/reztpl do my essay online] i implore you to visit the web site. There is customer care team that is available to the customers at any hour of the day. The main purpose of essay writing is to pass information to an identified audience in the society. The tips on essay writing will help students write quality essays that will earn them high marks in the essay papers. It does not have to be a formal outline, but just sketch out the order of your essay in some method that you understand. <br><br>Write a quick schedule to help you allocate your time wisely: list the days you have left and the time during the day you will be able to work. should always be the principal priority of a company. In essay writing, getting failing grades is the main fear of students but it is just an instigator of other apprehensions - as the writing process progresses, the main concern branches out to more precise fears. Our essay writing website is the best original essay writing service one can come across while searching for original papers, original essays, original term papers, original research papers, original academic papers, original thesis papers or original English essay writing. You should conclude the essay by focusing on the main purpose of the essay.
 
: <math> \chi: \Lambda_d\to\mathbf{T}, \quad
\chi\in\operatorname{Hom}(\Lambda_d,\mathbf{T}). </math>
 
A '''strong character''' is a restriction to ''&Lambda;'' of a continuous homomorphism from ''G'' to '''T''', that is an element of the [[Pontryagin dual]] of ''G''.
 
A set ''&Lambda;'' is '''harmonious''' if every weak character may be approximated by
strong characters uniformly on ''&Lambda;''. Thus for any ''&epsilon;'' > 0 and any weak character ''&chi;'', there exists a strong character ''&xi;'' such that
 
: <math> \sup_\Lambda |\chi(\lambda)-\xi(\lambda)| \leq \epsilon, \quad
\chi\in\operatorname{Hom}(\Lambda_d,\mathbf{T}), \xi\in\hat{G}. </math>
 
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 ''&Lambda;'' of ''G'' and a positive ''&epsilon;'', let ''M''<sub>''&epsilon;''</sub> be the subset of the Pontryagin dual of ''G'' consisting of all characters that are almost trivial on ''&Lambda;'':
 
: <math> \sup_\Lambda|\chi(\lambda)-1| \leq \epsilon, \quad
\chi\in\hat{G}.</math>
 
Then ''&Lambda;'' is '''harmonious''' if the sets ''M''<sub>''&epsilon;''</sub> are '''relatively dense''' in the sense of [[Besicovitch]]: for every ''&epsilon;'' > 0 there exists a compact subset ''K''<sub>''&epsilon;''</sub> of the Pontryagin dual such that
 
: <math> M_\epsilon + K_\epsilon = \hat{G}.</math>
 
== Properties ==
 
* A subset of a harmonious set is harmonious.  
 
* If ''&Lambda;'' is a harmonious set and ''F'' is a finite set then the set ''&Lambda;'' + ''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 ''&Lambda;'' 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.
 
== Examples ==
 
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 number]]s, ''&theta;'' >1, and the set ''&Lambda;'' consist of all finite sums of different powers of ''&theta;''. Then ''&Lambda;'' is harmonious if and only if ''&theta;'' 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 ''&Lambda;'' consist of all Pisot or [[Salem number|Salem]] numbers of degree ''n'' in '''K'''. Then ''&Lambda;'' is contained in the open interval (1,&infin;), 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''.
 
== See also ==
 
* [[Almost periodic function]]
 
== References ==
 
* [[Yves Meyer]], ''Algebraic numbers and harmonic analysis'', North-Holland Mathematical Library, vol.2, North-Holland, 1972
 
[[Category:Harmonic analysis]]
[[Category:Diophantine approximation]]
[[Category:Tessellation]]

Revision as of 08:03, 10 February 2014


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