## What is wrong with the proof of Theorem 1.39 of “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?

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See the comment of Fernando (November 9, 2010 7:27 pm) on that page. – Mark Lewko Sep 6 at 6:30