This paper is concerned with an important matrix condition in compressed sensing known as the restricted isometry property (RIP). We demonstrate that testing whether a matrix satisfies RIP is NP-hard. As a consequence of our result, it is impossible to efficiently test for RIP provided P = NP.

In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent ω of matrix multiplication by reducing matrix multiplication to group algebra multiplication, and in 2005 Cohn, Kleinberg, Szegedy, and Umans proposed specific conjectures for how to obtain ω = 2. In this paper we rule out obtaining ω = 2 in this framework from abelian groups of bounded exponent. To do this we bound the size of tricolored sum-free sets in such groups, extending the breakthrough results of Croot, Lev, Pach, Ellenberg, and Gijswijt on cap sets. As a byproduct of our proof, we show that a variant of tensor rank due to Tao gives a quantitative understanding of the notion of unstable tensor from geometric invariant theory.

We prove non-trivial bounds for general bilinear forms in hyper-Kloosterman sums when the sizes of both variables may be below the range controlled by Fourier-analytic methods (Pólya-Vinogradov range). We then derive applications to the second moment of cusp forms twisted by characters modulo primes, and to the distribution in arithmetic progressions to large moduli of certain Eisenstein-Hecke coefficients on GL3. Our main tools are new bounds for certain complete sums in three variables over finite fields, proved using methods from algebraic geometry, especially ℓ-adic cohomology and the Riemann Hypothesis.

Let G be an abelian group. A tri-colored sum-free set in G is a collection of triples (a a a i , b b b i , c c c i ) in G such that a a a i + b b b j + c c c k = 0 if and only if i = j = k. Fix a prime q and let C q be the cyclic group of order q. Let θ = min ρ>0 (1 + ρ + · · · + ρ q−1 )ρ −(q−1)/3 . Blasiak, Church, Cohn, Grochow, Naslund, Sawin, and Umans (building on previous work of Croot, Lev and Pach, and of Ellenberg and Gijswijt) showed that a tri-colored sum-free set in C n q has size at most 3θ n . Between this paper and a paper of Pebody, we will show that, for any δ > 0, and n sufficiently large, there are tri-colored sum-free sets in C n q of size (θ − δ ) n . Our construction also works when q is not prime. IntroductionLet G be an abelian group. Let t t t ∈ G n . We make the following slightly nonstandard definition: a sum-free set in G n with target t t t is a collection of triples (a a ato make the target 0 0 0 (as we did in the abstract, and as is more standard), but allowing an arbitrary target will simplify our notation. The usual terminology is "tri-colored sum-free set", but we omit the reference to the coloring as we never consider any other kind.If X ⊂ G n is a set with no three-term arithmetic progressions, then {(x x x, x x x, −2x x x) : x x x ∈ X} is sum-free with target 0 0 0, so lower bounds on sets without three-term arithmetic progressions are also bounds on sum-free sets. The reverse does not hold: the largest known three-term arithmetic progression free subsets of C n 3 (where C q is the cyclic group of order q) are of size 2.217 n [10]. Before this paper, the largest known sum-free sets in C n 3 were of size 2.519 n [1]; this paper will raise the bound to 2.755 n and show that this bound is tight. Letting r 3 (G n ) denote the largest subset of G n with no three-term arithmetic progressions, the question of whether lim sup n→∞ r 3 (G n ) 1/n < |G| was open, until recently, for every abelian G containing elements of order greater than two. The breakthrough work of Croot, Lev, and Pach [9] introduced a polynomial method to prove that strict inequality holds when G is cyclic of order 4, and Ellenberg and Gijswijt [12] built upon their ideas to prove it for cyclic groups of odd prime order. Blasiak et al. [5] applied the same method to prove upper bounds for sum-free sets in G n for any fixed finite abelian group G.We recall here one case of their bound. Let C q be the cyclic group of order q. Let θ = min β >0 (1 + β + · · · + β q−1 )β −(q−1)/3 and let ρ be the value of β at which the minimum is attained. We note that the minimum is attained at a unique point which belongs to (0, 1) because (1 + β + · · · + β q−1 )β −(q−1)/3 approaches ∞ as

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