Did Erdős publish his proof of the multiplicative version of the Erdős-Turán conjecture? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T00:36:40Z http://mathoverflow.net/feeds/question/49695 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49695/did-erds-publish-his-proof-of-the-multiplicative-version-of-the-erds-turan-conj Did Erdős publish his proof of the multiplicative version of the Erdős-Turán conjecture? Gjergji Zaimi 2010-12-17T01:47:36Z 2010-12-18T02:26:03Z <p>I read in an article of Erdős ("Extremal problems in number theory") that he had a proof of the multiplicative version of the <a href="http://mathoverflow.net/questions/43995/are-any-good-strategies-known-for-erdos-turan-conjecture-on-additive-bases-of-ord" rel="nofollow">Erdős-Turán conjecture</a>. The statement of this theorem is</p> <blockquote> <p>Let $a_1 &lt; a_2 &lt; \cdots$ and denote by $g(n)$ the number of solutions to $n=a_ia_j$. Then $g(n)>0$ for all $n>n_0$ implies $$\limsup_{n\to \infty} \ g(n)=\infty.$$</p> </blockquote> <p>He also claims that this follows under the weaker assumption: let $A(x)=\sum_{a_i &lt; x} 1$. Assume that for every $k$ we have $$\limsup_{x\to \infty} \ A(x)\left(x\left(\frac{\log\log x}{\log x}\right) ^k \right)^{-1}=\infty$$ than the same conclusion follows. However he also says that the proofs are difficult and haven't been published yet. </p> <p>Since that article is from 1965 I am assuming he must have published something about this theorem afterward, but I don't have a reference. I have seen a proof of the first statement before (not by Erdős), but not the second one. Does anyone know if these proofs were published, simplified or generalized?</p> http://mathoverflow.net/questions/49695/did-erds-publish-his-proof-of-the-multiplicative-version-of-the-erds-turan-conj/49760#49760 Answer by Mark Lewko for Did Erdős publish his proof of the multiplicative version of the Erdős-Turán conjecture? Mark Lewko 2010-12-17T22:32:22Z 2010-12-17T22:32:22Z <p>Yes.</p> <p>P. Erdõs: On the multiplicative representation of integers, Israel J. Math. 2 (1964), 251--261 (see: <a href="http://www.renyi.hu/~p_erdos/1964-20.pdf" rel="nofollow">http://www.renyi.hu/~p_erdos/1964-20.pdf</a> )</p> <p>A somewhat different proof is given in: </p> <p>Nešetřil, Rödl, Two proofs in combinatorial number theory. Proc. Amer. Math. Soc. 93 (1985), no. 1, 185–188. </p> http://mathoverflow.net/questions/49695/did-erds-publish-his-proof-of-the-multiplicative-version-of-the-erds-turan-conj/49769#49769 Answer by Mel Nathanson for Did Erdős publish his proof of the multiplicative version of the Erdős-Turán conjecture? Mel Nathanson 2010-12-18T02:26:03Z 2010-12-18T02:26:03Z <p>There is a generalization of Erdos' theorem in my paper "Multiplicative representations of integers," Israel Journal of Mathematics 57 (1987), 129--136.</p>