The following question is from Tao and Vu's Additive combinatorics, at which I am stuck and would appreciate some help.These questions are from the proof of Proposition 10.32 (page 392):
(Page 394) how does the followine equation follow from regularity of $\text{Bohr}(S,\rho)$: $\mathbb{E}_{x\in Z}f(x)1_{\text{Bohr}(S,\rho-\rho_1)}(x) \ge \frac{\delta}{2}\mathbb{P}_Z(\text{Bohr}(S,\rho))$. I am assuming this has to do something with definition 4.24 of regular bohr sets, but i don't see how.
(page 394) in the last para: how is the "crude" bound obtained from the boundedness of $f$ and regularity of $\text{Bohr}(S,\rho)$ :
$\mathbb{P}_z(\text{Bohr}(S,\rho+\rho_2))-\mathbb{P}_Z(\text{Bohr}(S,\rho))\le \frac{\delta^3}{8}\mathbb{P}_Z(\text{Bohr}(S,\rho-\rho_2))$ and then how does this crude bound allows restriction of $y$ to $\text{Bohr}(S,\rho-\rho_2)$ and how do we use the triangle inequality to obtain $$\mathbb{E}_{y\ \in \text{Bohr}(S,\rho-\rho_2)}F(y) \ge \frac{\delta^3}{8}$$
- (page 395) should the equation be $1_{\text{Bohr}(S,\rho_1)}(r)=1_{y+\text{Bohr}(S,\rho_1)}(y+z+r)1_{2\cdot\text{Bohr}(S,\rho_1)}(y+z+2r)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(*)\;\;$ instead of $1_{\text{Bohr}(S,\rho_1)}(r)-1_{y+\text{Bohr}(S,\rho_1)}(y+z+r)1_{2\cdot\text{Bohr}(S,\rho_1)}(y+z+2r)$, as in the equation below concerning $\tilde{F}(y)$, the expression $1_{\text{Bohr}(S,\rho_1)}(r)$ has been replaced by the RHS of the $(*)$.
If yes, can you please explain how we obtain the equation $(*)$.
Thanks.