Abstract — We present new combinatorial algorithms for Boolean matrix multiplication (BMM) and preprocessing a graph to answer independent set queries. We give the first asymptotic improvements on combinatorial algorithms for dense BMM in many years, improving on the “Four Russians ” O(n 3 /(w log n)) bound for machine models with wordsize w. (For a pointer machine, we can set w = log n.) The algorithms utilize notions from Regularity Lemmas for graphs in a novel way. • We give two randomized combinatorial algorithms for BMM. The first algorithm is essentially a reduction from BMM to the Triangle Removal Lemma. The best known bounds for the Triangle Removal Lemma only imply an O ` (n 3 log β)/(βw log n) ´ time algorithm for BMM where β = (log ∗ n) δ for some δ> 0, but improvements on the Triangle Removal Lemma would yield corresponding runtime improvements. The second algorithm applies the Weak Regularity Lemma of Frieze and Kannan along with “ several information compression ideas, running in O n 3 (log log n) 2 /(log n) 9/4 ”) time with probability exponentially “ close to 1. When w ≥ log n, it can be implemented in O n 3 (log log n) 2 /(w log n) 7/6 ”) time. Our results immediately imply improved combinatorial methods for CFG parsing, detecting triangle-freeness, and transitive closure. Using Weak Regularity, we also give an algorithm for answering queries of the form is S ⊆ V an independent set? in a graph. Improving on prior work, we show how to randomly preprocess a graph in O(n 2+ε) time (for all ε> 0) so that with high probability, all subsequent batches of log n independent “ set queries can be answered deterministically in O n 2 (log log n) 2 /((log n) 5/4 ”) time. When w ≥ log n, w queries can be answered in O n 2 (log log n) 2 /((log n) 7/6 ” time. In addition to its nice applications, this problem is interesting in that it is not known how to do better than O(n 2) using “algebraic ” methods. 1.

In this paper we study the structure of polynomials of degree three and four that have high bias or high Gowers norm, over arbitrary prime fields. In particular we obtain the following results. 1. Let f be a degree three polynomial with bias(f) = δ then there exist r = O(log(1/δ)) quadratic polynomials {qi}, c = O(log 4 ( 1 δ)) linear functions {ℓi} and a degree three polynomial g such that f = ∑r i=1 ℓi · qi + g(ℓ1,..., ℓc). This result generalizes the corresponding result for quadratic polynomials. 2. Let deg(f) = 4 and bias(f) = δ. Then f = ∑r i=1 ℓi · gi + ∑r i=1 qi · q ′ i, where r = poly(1/δ), the ℓi-s are linear, the qi-s are quadratics and the gi-s are cubic. 3. Let deg(f) = 4 and ‖f ‖ U 4 = δ. Then there exists a partition of a subspace V ⊆ F n, dim(V) ≥ n − O(log(1/δ)), to subspaces {Vα}, such that ∀α dim(Vα) ≥ n / exp(log 2 (1/δ)) and deg(f|Vα) = 3. Items 1,2 extend and improve previous results for degree three and four polynomials [KL08, GT07]. Item 3 gives a new result for the case of degree four polynomials with high U 4 norm. It is the first case where the inverse conjecture for the Gowers norm fails [LMS08, GT07], namely that such an f is not necessarily correlated with a cubic polynomial. Our result shows that instead f equals a cubic polynomial on a large subspace (in fact we show that a much stronger claim holds). Our techniques are based on finding a structure in the space of partial derivatives of f. For example, when deg(f) = 4 and f has high U 4 norm we show that there exist quadratic polynomials {qi} i∈[r] and linear functions {ℓi} i∈[R] such that (on a large enough subspace) every partial derivative of f can be written as ∆y(f) = ∑R i=1 ℓi · q y i + ∑r i=1 qi · ℓ y i + qy 0, where ℓy

Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define – perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which are used there. One might say that additive combinatorics is a branch of mathematics concerning the study of additive structures in sets equipped with a group structure – we may have other structure that interacts with this group structure. This newly emerging field has seen tremendous advances over the last few years, and has recently become a focus of attention among both mathematicians and computer scientists. This fascinating area has been enriched by its formidable links to combinatorics, number theory, harmonic analysis, ergodic theory, and some other branches; all deeply cross-fertilize each other, holding great promise for all of them! There is a considerable number of incredible problems, results, and novel applications in this thriving area. In this exposition, we attempt to provide an illuminating overview of some conspicuous breakthroughs in this captivating field, together with a number of seminal applications to sundry parts of mathematics and some other disciplines, with emphasis on computer science and cryptography.