TITLE: What makes a set random?
QUESTION [6 upvotes]: There are many results in number theory, where the existence
of some $B \subseteq \mathbb{N}$ with certain properties is proved by
a probabilistic argument employing "random sets". One such example
would be the result of Erdős and Rényi, where they proved the existence
of a "thin" $B_2[g]$ sequence.
My question is are there certain known properties, so that if $A \subseteq \mathbb{N}$
satisfies them, then $A$ is considered to behave like a random set? or what properties would make $A$ considered to be close to being random? Thank you!

REPLY [2 votes]: Additive number theory (or additive combinatorics) makes a lot of use of quasirandom sets.  
Suppose that we are looking for an arithmetic progression in some dense subset $A$ of $[n] = \{1, \ldots, n\}$.  If $A$ resembles a random set, then finding an AP is easy, so $A$ must have some structure.  That structure turns out to be that $A$ is slightly denser on some subprogression of $A$ (e.g. $\{2, 4, 6, \ldots, n\}$, so we can repeat the argument on this subset with a stronger assumption.  At a very high level, this is one way that Szemerédi's theorem can be proved.
So one way of characterising quasirandom sets would be to say that they have about the density you would expect on all subprogressions.  This property can be quantified using various analytic tools: for example, we might say that a set is quasirandom if (most of) the fourier coefficients of its indicator function are small. 
Tao and Vu's book has already been recommended to you, and would have a lot more detail on this sort of thing.