As corollaries, we obtain:
(1) Starting from a PCP Theorem with soundness error bounded away from 1, we get a PCP with arbitrarily small constant soundness error. In particular, starting with the combinatorial PCP of Dinur, we get a combinatorial PCP with low error. The latter can be used for hardness of approximation as in the work of Hastad.
(2) Starting from the work of the author and Raz, we get a projection PCP theorem with the smallest soundness error known today. The theorem yields nearly a quadratic improvement in the size compared to previous work.