Additive Combinatorics mini course - errata by Rani Hod
... subsets into intersections of $ V_i, V_i \setminus S_i, V_j, V_j \setminus T_j$ over all non-regular ...
should be
... subsets into intersections of $S_i, V_i \setminus S_i, T_j, V_j \setminus T_j$ over all non-regular ...
- In section 3.1.1, Theorem 3.1.4:
... The best $\epsilon$ known ... is 4/3, but it is conjectured ... for $\epsilon = 2 - o(1)$.
should be
... The best $\epsilon$ known ... is 1/3, but it is conjectured ... for $\epsilon = 1 - o(1)$.
... a condition that $\mathbb{F}$ contains on subfields.
should be
... a condition that $\mathbb{F}$ contains no subfields.
- In section 3.1.2, Theorem 3.1.5:
... and it is known that $\epsilon$ cannot be larger than 3/2.
should be
... and it is known that $\epsilon$ cannot be larger than 1/2.
- In section 3.2.1, definition of $I$:
$$ I = \{ ... such that point p lies on line l \} $$
should be
$$ I = \{ ... such that point $p$ lies on line $l$ \} $$
... but the same procedure may be plied to the other three ...
should be
... but the same procedure may be applied to the other three ...
- In section 3.2.2, just after Theorem 3.2.7:
This is essentially Theorem 3.2.1; maybe it's better to prove the more general case in section 3.2.1.
Also, maybe refrain from using $p$ to denote both the field characteristic and a member of $P$.
- In section 3.2.3, Definition 3.2.8:
$\omega$ is not defined. I reckon it is the (complex) $|G|$-th root of unity.
- In section 3.2.5, Definition 3.2.14:
$$ || f(x) - U_m ||_1 \le \epsilon$$ is an $(S, \epsilon)$-disperser.
should be
$$ || f(X) - U_m ||_1 \le \epsilon$$ is an $(S, \epsilon)$-extractor.
- In section 3.2.5, last paragraph before "A Statistical Version of ...":
... Before the sum-product theorem, Erdos (using the ...
should be
... Before the sum-product theorem, Erd\"{o}s (using the ...
- In section 3.2.5, just before Definition 3.2.18:
... for $k = \delta n$ and $c = \textrm{poly}(1/\delta)$ Note that ...
should be
... for $k = \delta n$ and $c = \textrm{poly}(1/\delta)$. Note that ...
- In section 3.2.5, Definition 3.2.18:
$$ \exists c > 0 : ... $$
is $c$ dependent on $X$? on $\epsilon$? should $f_c$ be a family of functions defined for many values of $c$?
- In section 3.3, proof of Lemma 3.3.2:
... It is left to prove the claim: assume otherwise, ie that $\delta' < \frac 1k$.
should be
... It is left to prove the claim: assume otherwise, that is, $\delta' < \frac 1k$.
("ie that" appears once more just after Theorem 3.3.6.)
- In section 3.3, proof of Lemma 3.3.2:
... such that each $s_j$ satisfies $s_j \not\in s_0 A' + s_1 A' + \ldots s_{j-1} A'$.
should be
... such that each $s_j$ satisfies $s_j \not\in s_0 A' + s_1 A' + \cdots + s_{j-1} A'$.
(The same expression appears two more times throughout the proof.)
- In section 3.3, Theorem 3.3.6:
$||A + A||^{-1} < |A|^{1+\epsilon}$
should be (?)
$|A + A|^{-1} < |A|^{1+\epsilon}$
- In section 3.3, after Theorem 3.3.6:
(i) Lemma 1 and Lemma 2 should be \ref{}s to Lemma 3.3.2 and 3.3.3, respectively.
(ii) The contrapositive assumptions
$|A+A| \le |A|^{1+\epsilon}$ and $|A\times A| \le |A|^{1+\epsilon}$
should be strict
$|A+A| < |A|^{1+\epsilon}$ and $|A\times A| < |A|^{1+\epsilon}$
(iii) The last sentence in the bracketed proof is missing end punctuation.