Bounds on size of subset of additive set that is sum-free

[poissonspot]

Hey, I was reading about this is in "Additive Combinatorics" (Additive Combinatorics (Cambridge Studies in Advanced Mathematics): Terence Tao, Van H. Vu: 9780521136563: Amazon.com: Books) on pg. 4 when I went to the references and found this paper (http://renyi.mta.hu/~p_erdos/1965-02.pdf) which proves the statement starting at the bottom of the 6th page. The proofs look very different, but I feel like the same ideas might be being developed. A couple questions:

In Tao's proof, I'm not sure why $x^{-1}a$ is uniformly distributed in Z_p\{0}

In Erdos's, I am unfamiliar with this notation, "a_r,alpha (mod 1)".

Any suggestions? Thx

 

[Valkarie]

!In Tao's proof, the reason why $x^{-1}a$ is uniformly distributed in Z_p\{0} is because of the properties of the inverse map. In particular, if we let $f:Z_p \rightarrow Z_p$ be defined by $f(x) = x^{-1}a$, then for any $y \in Z_p\{0\}$, we have $f^{-1}(y) = yx$, which implies that $f^{-1}$ is bijective, and thus $f$ is also bijective. This means that $f$ is a 1-1 onto map, and hence $x^{-1}a$ must be uniformly distributed over $Z_p\{0\}$.In Erdos's paper, "a_r,alpha (mod 1)" refers to a sequence of points on the unit circle, where $a_r$ is the $r$th point in the sequence and $\alpha$ is an angle. Specifically, the sequence is defined as follows: $a_0 = 0, a_1 = e^{2\pi i \alpha}, a_2 = e^{4\pi i \alpha},...,a_r = e^{2r\pi i \alpha}$.