Abstract
We give a new explicit construction of n x N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some e>0, large k and k2-e ≤ N ≤ k2+e, we construct RIP matrices of order k with n=O(k2-e). This overcomes the natural barrier n >> k2 for proofs based on small coherence, which are used in all previous explicit constructions of RIP matrices. Key ingredients in our proof are new estimates for sumsets in product sets and for exponential sums with the products of sets possessing special additive structure.
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