An additive combinatorics approach to the log-rank conjecture in communication complexity

For a {0, 1}-valued matrixM let CC(M) denote the deterministic communication complexity of the boolean function associated with M . The log-rank conjecture of Lovász and Saks [FOCS 1988] states that CC(M) ≤ logc(rank(M)) for some absolute constant c where rank(M) denotes the rank of M over the field of real numbers. We show that CC(M) ≤ c · rank(M)/ log rank(M) for some absolute constant c, assuming a well-known conjecture from additive combinatorics known as the Polynomial Freiman-Ruzsa (PFR) conjecture. Our proof is based on the study of the “approximate duality conjecture” which was recently suggested by Ben-Sasson and Zewi [STOC 2011] and studied there in connection to the PFR conjecture. First we improve the bounds on approximate duality assuming the PFR conjecture. Then we use the approximate duality conjecture (with improved bounds) to get the aforementioned upper bound on the communication complexity of low-rank martices, where this part uses the methodology suggested by Nisan and Wigderson [Combinatorica 1995].