Gowers uniformity, influence of variables, and PCPs

We return to the study of the relation of query complexity and soundness in probabilistically checkable proofs. We present a PCP verifier for languages that are Unique-Games-Hard and such that the verifier makes q queries, has almost perfect completeness, and has soundness error at most 2q/2^q + ε, for arbitrarily small ε > 0. For values of q of the form 2 ^t - 1, the soundness error is (q + 1)/2^q + ε. Charikar et al. show that there is a constant c such that for every language that has a verifier of query complexity q, and a ratio of soundness error to completeness smaller than cg/2^q is decidable in polynomial time. Up to the value of the multiplicative constant and to the validity of the Unique Games Conjecture, our result is therefore tight. As a corollary, we show that approximating the Maximum Independent Set problem in graphs of degree A within a factor better than Δ/(log δ)^c is Unique-Games-Hard for a certain constant c > 0. Our main technical results are (i) a connection between the Gowers uniformity of a Boolean function and the influence of its variables and (ii) the proof that "Gowers uniform" functions pass the "hypergraph linearity test" approximately with the same probability of a random function. The connection between Gowers uniformity and influence might have other applications.