What is wrong with the proof of Theorem 1.39 of "Additive Combinatorics"? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T09:46:59Z http://mathoverflow.net/feeds/question/106479 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106479/what-is-wrong-with-the-proof-of-theorem-1-39-of-additive-combinatorics What is wrong with the proof of Theorem 1.39 of "Additive Combinatorics"? unknown (google) 2012-09-06T05:50:54Z 2012-09-06T05:50:54Z <p>According to <a href="http://terrytao.wordpress.com/books/additive-combinatorics/" rel="nofollow">http://terrytao.wordpress.com/books/additive-combinatorics/</a></p> <p>we have:</p> <p>p. 36-37: The proof of Theorem 1.39 requires a number of significant changes. After the first paragraph, add “We will show first that with probability 1, that any natural number has at most a bounded number of representations as the sum of $k$ elements of $A$ between $n^\epsilon$ and $n$; the treatment of the remaining sums in which at least one term is less than $n^\epsilon$ is left as an exercise.” ...</p> <p>Now, this is referring to Theorem 1.39 of Additive Combinatorics:</p> <p>For any $h \geq 1$ and $\epsilon > 0$, there exists a set $A \subset Z^+$ with $|A \cap [0,n]| = \Omega_h(n^{1/h-\epsilon})$ for all large $n$, which is a $B_h[g]$ set for some $g=g_{h, \epsilon}$.</p> <p>Now, the part of the proof which I think the errata above mentions is:</p> <p>$E(Y_m) = \sum_{n_1 \leq ... \leq n_h: n_1 + ... +n_h = m} n_1^{1/h - \epsilon} ... n_h^{1/h-\epsilon}$= ...</p> <p>Now, supposedly there is an error here (and thus the need for the errata). What is wrong with the proof of Theorem 1.39?</p>