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For every e > 0 and integers k and q (with k < q <_ k+k2/4), we present a PCP characterization of NP, where the verifier queries q bits (of which only k are free bits), accepts a correct proof with probability >__ 1 -e and accepts a "proof" of a wrong statement with probability ~ 2 -(q-k). In particular, for every 6 > 0 we have a PCP characterization of NP, where the verifier has, simultaneously, 1 + 5 amortized query complexity and 6 amortized free bit complexity. Both results are tight, unless P ----NP.The optimal amortized query complexity of our verifier implies essentially tight non-approximability results for constraint satisfaction problems. Specifically, we can show that k-CSP, the problem of finding an assignment satisfying the maximum number of given constraints (where each constraint involves at most k variables) is NP-hard to approximate to within a factor 2 -~+°(v~). The problem can be approximated to within a factor 2 -~+1, and was known to be NP-hard to approximate to within a factor about 2 -2k/3. We can also prove some new separation results between different PCP models. A PCP characterization of NP with optimal amortized free bit complexity implies that for every 5 > 0 it is hard to approximate the maximum clique problem to within a n 1-~ factor. Such a characterization had already been proved by H~stad [13], in a celebrated recent breakthrough. Our construction gives an alternative, simpler, proof of this result.Our techniques also give a tight analysis of linearity testing algorithms with low amortized query complexity. As in the case of PCP, we show that it is possible to have a linearity testing algorithm that makes q queries and has error bounded from above by 2 -q+°(x/~). We also prove a lower bound showing that, for a certain, fairly general, class of testing algorithms, our analysis is tight even in the lower order term. That is, we show that the error of a q-query testing algorithm in this class has to be at least 2 -q+~(C4)

We define tests of boolean functions which distinguish between linear (or quadratic) polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal trade-offs between soundness and the number of queries.In particular, we show that functions with small Gowers uniformity norms behave "randomly" with respect to hypergraph linearity tests.A central step in our analysis of quadraticity tests is the proof of an inverse theorem for the third Gowers uniformity norm of boolean functions.The last result has also a coding theory application. It is possible to estimate efficiently the distance from the second-order Reed-Muller code on inputs lying far beyond its listdecoding radius.

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