Quasirandomness and Regularity: Lecture Notes Iv

One of the most beautiful, and useful, areas in which quasirandomness has been studied concerns the subsets of the (rational) integers modulo n, which we write Zn. We need a few items of notation before introducing the random-like properties we are interested in. First of all, for x ∈ Zn, let en(x) = e. When n is understood, we simply write e(x). Then, for a subset S ⊂ Zn, let the function χS : Zn → {0, 1} denote the characteristic function of S, i.e., χS(x) = 0 if x 6∈ S and 1 otherwise. Also, S + t = {s + t : s ∈ S}, and #(S ⊂ T ) = |{x : S + x ⊂ T}|, or, informally, the “number of copies of S in T .” Note that #(S ⊂ T ) = | ⋂ u∈S(T − u)|. Finally, we define GS to be the graph with vertex set Zn so that {i, j} ∈ E(GS) iff i + j ∈ S. The following are our quasirandom properties for the “ambient set” S ⊂ Zn with cardinality s = |S|. Where a “test set” T is involved, t denotes the cardinality of T .