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| In [[number theory]], a '''Sidon sequence''' (or '''Sidon set'''), named after the Hungarian mathematician [[Simon Sidon]], is a sequence ''A'' = {''a''<sub>0</sub>, ''a''<sub>1</sub>, ''a''<sub>2</sub>, ...} of natural numbers in which all pairwise sums ''a''<sub>''i''</sub> + ''a''<sub>''j''</sub> (''i'' ≤ ''j'') are different. Sidon introduced the concept in his investigations of [[Fourier series]].
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| The main problem in the study of Sidon sequences, posed by Sidon,<ref>{{citation|first1=P.|last1=Erdős|author1-link=Paul Erdős|first2=P.|last2=Turán|author2-link=Pál Turán|title=On a problem of Sidon in additive number theory and on some related problems|journal=J. London Math. Soc.|volume=16|year=1941|pages=212–215|doi=10.1112/jlms/s1-16.4.212|url=http://www.renyi.hu/~p_erdos/1941-01.pdf}}. [http://www.math-inst.hu/~p_erdos/1944-02.pdf Addendum], '''19''' (1944), 208.</ref> is to find the largest number of elements a Sidon sequence ''A'' can have smaller than some given number ''x''. Despite a large body of research,<ref>{{citation|first=K.|last=O'Bryant|url=http://www.emis.ams.org/journals/EJC/Surveys/ds11.pdf|title=A complete annotated bibliography of work related to Sidon sequences|journal=Electronic Journal of Combinatorics|volume=11|year=2004|page=39}}.</ref> the question remained unsolved for almost 80 years. In 2010, it was finally settled<ref>{{citation|first1=J.|last1=Cilleruelo|author1-link=Javier Cilleruelo|first2=I.|last2= Ruzsa|author2-link=Imre Z. Ruzsa|first3=C.|last3=Vinuesa|author3-link=Carlos Vinuesa|title=Generalized Sidon sets|journal=Advances in Mathematics|volume=225|year=2010|pages=2786–2807|url=http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6W9F-505G2M9-1-1&_cdi=6681&_user=1675225&_pii=S0001870810001945&_origin=gateway&_coverDate=12%2F01%2F2010&_sk=997749994&view=c&wchp=dGLzVtb-zSkWb&md5=e3adbe497603d81442ebb8f745334fc4&ie=/sdarticle.pdf}}</ref> by J. Cilleruelo, [[Imre Z. Ruzsa|I. Ruzsa]] and C. Vinuesa.
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| ==Early results==
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| [[Paul Erdős]] and [[Pál Turán]] proved that, for every ''x'' > 0, the number of elements smaller than ''x'' in a Sidon sequence is at most <math>\sqrt{x}+O(\sqrt[4]{x})</math>. Using a construction of J. Singer, they showed that there exist Sidon sequences that contain <math>\sqrt{x}(1-o(1))</math> terms less than ''x''.
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| ==Infinite Sidon sequences==
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| Erdős also showed that if we consider any particular infinite Sidon sequence ''A'' and let ''A''(''x'') denote the number of its elements up to ''x'', then
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| : <math>\liminf_{x \to \infty} \frac{A(x)\sqrt{\log x}}{\sqrt{x}}\leq 1</math>.
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| That is, infinite Sidon sequences are thinner than the densest finite Sidon sequences.
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| For the other direction, [[Sarvadaman Chowla|Chowla]] and Mian observed that the greedy algorithm gives an infinite Sidon sequence with <math>A(x)>c\sqrt[3]{x}</math> for every ''x''.<ref>{{citation
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| | last1 = Mian | first1 = Abdul Majid
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| | last2 = Chowla | first2 = S. | author2-link = Sarvadaman Chowla
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| | journal = Proc. Nat. Acad. Sci. India. Sect. A.
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| | mr = 0014114
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| | pages = 3–4
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| | title = On the ''B''<sub>2</sub> sequences of Sidon
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| | volume = 14
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| | year = 1944}}.</ref> [[Miklós Ajtai|Ajtai]], [[János Komlós (mathematician)|Komlós]], and [[Endre Szemerédi|Szemerédi]] improved this with a construction<ref>{{citation|first1=M.|last1=Ajtai|author1-link=Miklós Ajtai|first2=J.|last2=Komlós|author2-link=János Komlós (mathematician)|first3=E.|last3=Szemerédi|author3-link=Endre Szemerédi|title=A dense infinite Sidon sequence|journal=European Journal of Combinatorics|volume=2|year=1981|pages=1–11|mr=0611925|issue=1}}.</ref> of a Sidon sequence with
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| : <math>A(x)>\sqrt[3]{x\log x}.</math> | |
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| The best lower bound to date was given by [[Imre Z. Ruzsa]], who proved<ref>{{citation|first=I. Z.|last=Ruzsa|authorlink=Imre Z. Ruzsa|title=An infinite Sidon sequence|journal=Journal of Number Theory|volume=68|year=1998|pages=63–71|mr=1492889|doi=10.1006/jnth.1997.2192}}.</ref> that a Sidon sequence with
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| : <math>A(x)>x^{\sqrt{2}-1-o(1)}</math>
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| exists. Erdős conjectured that an infinite Sidon set ''A'' exists for which <math>A(x)>x^{1/2-o(1)}</math> holds. He and [[Alfréd Rényi|Rényi]] showed<ref>{{citation|first1=P.|last1=Erdős|author1-link=Paul Erdős|first2=A.|last2=Rényi|author2-link=Alfréd Rényi|title=Additive properties of random sequences of positive integers|journal=Acta Arithmetica|volume=6|year=1960|pages=83–110|mr=0120213|url=http://www.renyi.hu/~p_erdos/1960-02.pdf}}.</ref> the existence of a sequence {''a''<sub>0</sub>,''a''<sub>1</sub>,...} with the conjectural density but satisfying only the weaker property that there is a constant ''k'' such that for every natural number ''n'' there are at most ''k'' solutions of the equation ''a''<sub>''i''</sub> + ''a''<sub>''j''</sub> = ''n''. (To be a Sidon sequence would require that ''k'' = 1.)
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| Erdős further conjectured that there exists a nonconstant [[integer]]-[[coefficient]] [[polynomial]] whose values at the [[natural numbers]] form a Sidon sequence. Specifically, he asked if the set of fifth powers is a Sidon set. Ruzsa came close to this by showing that there is a real number ''c'' with 0 < ''c'' < 1 such that the range of the function
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| ''f''(''x'') = ''x''<sup>5</sup> + [''cx''<sup>4</sup>] is a Sidon sequence, where [.] denotes [[integer part]]. As ''c'' is irrational, this function ''f''(''x'') is not a polynomial. The statement that the set of fifth powers is a Sidon set is a special case of the later conjecture of [[Euler's_sum_of_powers_conjecture#Generalizations|Lander, Parkin and Selfridge]].
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| ==Relationship to Golomb rulers==
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| All finite Sidon sets are [[Golomb ruler]]s, and vice-versa.
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| To see this, suppose for a [[proof by contradiction|contradiction]] that ''S'' is a Sidon set and not a Golomb ruler. Since it is not a Golomb ruler, there must be four members such that <math>a_i-a_j=a_k-a_l</math>. It follows that <math>a_i+a_l=a_k+a_j</math>, which contradicts the proposition that ''S'' is a Sidon set. Therefore all Sidon sets must be Golomb rulers. By a similar argument, all Golomb rulers must be Sidon sets.
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| ==See also==
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| *[[Sumset]]
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| ==References==
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| {{reflist|30em}}
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| * {{cite book |last=Guy |first=Richard K. |authorlink=Richard K. Guy |title=Unsolved problems in number theory |publisher=[[Springer-Verlag]] |edition=3rd |year=2004 |isbn=0-387-20860-7 |at=C9 |zbl=1058.11001}}
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| {{DEFAULTSORT:Sidon Sequence}}
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| [[Category:Number theory]]
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| [[Category:Combinatorics]]
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