  For a $\{0,1\}$-valued matrix $M$ let $\rm{CC}(M)$ denote the deterministic
communication complexity of the boolean function associated with $M$. The
log-rank conjecture of Lov\'{a}sz and Saks [FOCS 1988] states that $\rm{CC}(M)
\leq \log^c(\rm{rank}(M))$ for some absolute constant $c$ where $\rm{rank}(M)$
denotes the rank of $M$ over the field of real numbers. We show that
$\rm{CC}(M)\leq c \cdot \rm{rank}(M)/\log \rm{rank}(M)$ 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].
