Gowers Uniformity, Influence of Variables, and Pcps
Gowers Uniformity, Influence of Variables, and PCPs Alex Samorodnitsky∗ Luca Trevisany October 12, 2005 Abstract Gowers [Gow98, Gow01] introduced, for d 1, the notion of dimension-d uniformity U d(f) of a function f : G C, where G is a finite ab≥elian group. Roughly speaking, if a function has small Gowers uniformit! y 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 × · · · × d groups. If a bounded balanced function f : G1 Gn C is such that U (f) ", then one of the coordinates of f has influence at least×"=· ·2·O(d). !Other inverse theorems are≥ known [Gow98, Gow01, GT05, Sam05], and U 3 is especially well understood, but the properties of functions f with large U d(f), d 4, are not yet well characterized. ≥ The dimension-d Gowers inner product fS d of a collection fS of functions is a hf giU f gS⊆[d] related measure of pseudorandomness. The definition is such that if all the functions fS are d equal to the same fixed function f, then fS d = U (f). hf giU We prove that if fS : G Gn C is a collection of bounded functions such that 1 × · · · × ! fS U d " and at least one of the fS is balanced, then there is a variable that has influence atjhfleastgi "2j=≥2O(d) for at least four functions in the collection. Finally, we relate the acceptance probability of the \hypergraph long-code test" proposed by Samorodnitsky and Trevisan to the Gowers inner product of the functions being tested and we deduce the following result: if the Unique Games Conjecture is true, then for every q 3 there is a PCP characterization of NP where the verifier makes q queries, has almost perfect≥ completeness, and soundness at most 2q=2q.
[Show full text]