What is wrong with the proof of Theorem 1.39 of "Additive Combinatorics"?

2012-09-06T05:50:54Z

According to http://terrytao.wordpress.com/books/additive-combinatorics/

we have:

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.” ...

Now, this is referring to Theorem 1.39 of Additive Combinatorics:

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 $g=g_{h, \epsilon}$.

Now, the part of the proof which I think the errata above mentions is:

$E(Y_m) = \sum_{n_1 \leq ... \leq n_h: n_1 + ... +n_h = m} n_1^{1/h - \epsilon} ... n_h^{1/h-\epsilon}$= ...

Now, supposedly there is an error here (and thus the need for the errata). What is wrong with the proof of Theorem 1.39?

What is wrong with the proof of Theorem 1.39 of "Additive Combinatorics"? - MathOverflow

What is wrong with the proof of Theorem 1.39 of "Additive Combinatorics"?