Gowers uniformity, influence of variables, and PCPs

Gowers [Gow98, Gow01] introduced, for d 1, the notion of dimension-d uniformity Ud (f) of a function f : G C, where G is a finite abelian group. Roughly speaking, if a function has small Gowers uniformity of dimension d, then it "looks random" on certain structured subsets of the inputs. We prove the following "inverse theorem." Write G = G1 ? ? ? ? ? Gn as a product of groups. If a bounded balanced function f : G1 ? ? ? ? Gn C is such that Ud (f) , then one of the coordinates of f has influence at least /2O(d) . Other inverse theorems are known [Gow98, Gow01, GT05, Sam05], and U3 is especially well understood, but the properties of functions f with large Ud (f), d 4, are not yet well characterized. The dimension-d Gowers inner product {fS} Ud of a collection {fS}S[d] of functions is a related measure of pseudorandomness. The definition is such that if all the functions fS are equal to the same fixed function f, then {fS} Ud = Ud (f). We prove that if fS : G1 ...

• Type: Conference
• Year: 2006
• Where: STOC
• Authors: Alex Samorodnitsky, Luca Trevisan
• Keywords: Algorithms, Dimension-d Uniformity Ud, Gowers Inner Product, Product {fs} Ud, STOC 2006