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| The '''Erdős–Turán conjecture''' is an old unsolved problem in [[additive number theory]] (not to be confused with [[Erdős conjecture on arithmetic progressions]]) posed by Paul Erdős and Pál Turán in 1941.
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| ==History==
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| The conjecture was made jointly by [[Paul Erdős]] and [[Pál Turán]] in.<ref>{{Cite journal |last=Erdős |first=Paul. |last2=Turán|first2=Pál |year=1941 |title=On a problem of Sidon in additive number theory, and on some related problems |journal=Journal of the London Mathematical Society |volume=16 |pages=212–216 |doi=10.1112/jlms/s1-16.4.212}}</ref> In the original paper, they state
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| "(2) If <math>f(n) > 0 </math> for <math>n > n_0 </math>, then <math> \varlimsup_{n \rightarrow \infty} f(n) = \infty </math>"
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| Here <math>f(n)</math> is the number of ways one can write the natural number <math>n</math> as the sum of two (not necessarily distinct) elements of <math>B</math>. If <math>f(n)</math> is always positive for sufficiently large <math>n</math>, then <math>B</math> is called an additive basis (of order 2).<ref name="TaoVu13">{{cite book |authorlink=Terence Tao |first=T. |last=Tao |first2=V. |last2=Vu |title=Additive Combinatorics |location=New York |publisher=Cambridge University Press |year=2006 |isbn=0-521-85386-9 |page=13 }}</ref> This problem has attracted significant attention<ref name="TaoVu13" /> but remains unsolved.
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| In 1964, Erdős published a multiplicative version this conjecture. See source :
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| *P. Erdõs: On the multiplicative representation of integers, Israel J. Math. 2 (1964), 251--261
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| ==Progress==
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| While the conjecture remains unsolved, there have been significant advance on the problem. First, we express the problem in modern language. For a given subset <math>B \subset \mathbb{N} </math>, we define its ''representation function'' <math>r_B(n) = \#\{(a_1, a_2) \in B^2 | a_1 + a_2 = n \}</math>. Then the conjecture states that if <math> r_B(n) > 0 </math> for all <math>n</math> sufficiently large, then <math> \limsup_{n \rightarrow \infty} r_B(n) = \infty </math>.
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| More generally, for any <math>h \in \mathbb{N}</math> and subset <math>B \subset \mathbb{N} </math>, we can define the <math>h</math> representation function as <math>r_{B,h}(n) = \#\{(a_1, \cdots, a_h) \in B^h | a_1 + \cdots + a_h = n \}</math>. We say that <math>B</math> is an additive basis of order <math>h</math> if <math>r_{B,h}(n) > 0 </math> for all <math> n </math> sufficiently large. One can see from an elementary argument that if <math>B</math> is an additive basis of order <math>h</math>, then
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| <math> \displaystyle n \leq \sum_{m=1}^n r_{B,h}(m) \leq |B \cap [1,n]|^h </math>
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| So we obtain the lower bound <math> n^{1/h} \leq |B \cap [1,n]| </math>.
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| The original conjecture spawned as Erdős and Turán sought a partial answer to Sidon's problem (see: [[Sidon sequence]]). Later, Erdős set out to answer the following question posed by Sidon: how close to the lower bound <math> |B \cap [1,n]| \geq n^{1/h} </math> can an additive basis <math> B </math> of order <math> h </math> get? This question was answered positively in the case <math>h=2</math> by Erdős in 1956.<ref>{{cite journal |first=P. |last=Erdős |title=Problems and results in additive number theory |journal=Colloque sur le Theorie des Nombres |year=1956 |volume= |issue= |pages=127–137 |doi= }}</ref> Erdős proved that there exists an additive basis <math>B</math> of order 2 and constants <math>c_1, c_2 > 0 </math> such that <math>c_1 \log n \leq r_B(n) \leq c_2 \log n </math> for all <math>n </math> sufficiently large. In particular, this implies that there exists an additive basis <math>B</math> such that <math>r_B(n) = n^{1/2 + o(1)} </math>, which is essentially best possible. This motivated Erdős to make the following conjecture
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| If <math>B</math> is an additive basis of order <math>h</math>, then <math> \limsup_{n \rightarrow \infty} r_B(n)/\log n > 0.</math>
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| In 1986, [[Eduard Wirsing]] proved that a large class of additive bases, including the [[prime numbers]], contains a subset that is an additive basis but significantly thinner than the original.<ref>{{Cite journal |last = Wirsing |first=Eduard | year = 1986 | title = Thin subbases |journal=Analysis | volume=6 |issue= | pages= 285–308 }}</ref> In 1990, Erdős and Tetalli extended Erdős's 1956 result to bases of arbitrary order.<ref>{{Cite journal |last=Erdős |first=Paul. |last2=Tetalli|first2=Prasad |year=1990 |title=Representations of integers as the sum of <math>k</math> terms|journal=Random Structures Algorithms|volume=1 |issue = 3 |pages=245–261 |doi=10.1002/rsa.3240010302 }}</ref> In 2000, [[Van H. Vu|V. Vu]] proved that thin subbases exist in the Waring bases using the [[Hardy–Littlewood circle method]] and his polynomial concentration results.<ref>{{Cite journal |last = Vu | first=Van | year = 2000 | title = On a refinement of Waring's problem |journal=Duke Mathematical Journal |volume=105 | issue = 1 |pages= 107–134 |doi=10.1215/S0012-7094-00-10516-9 }}</ref> In 2006, Borwein, Choi, and Chu proved that for all additive bases <math>B</math>, <math>f(n)</math> eventually exceeds 7.<ref>{{Cite journal |last = Borwein | first=Peter | last2 = Choi | first2=Stephen |last3=Chu |first3=Frank|year=2006|title = An old conjecture of Erdős–Turán on additive bases |journal=Mathematics of Computation |volume=75 |issue= |pages=475–484 |doi= }}</ref>
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| <ref>{{cite book |last1= Xiao |first1= Stanley Yao |title= On the Erdős–Turán conjecture and related results|url= http://hdl.handle.net/10012/6150 |degree= masters |year = 2011 }}</ref>
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Erdos-Turan Conjecture On Additive Bases}}
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| [[Category:Additive number theory]]
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Got nothing to write about myself at all.
Great to be a part of this site.
I really wish I'm useful at all
Check out my web page: tre khoe dep