Let $f: {0,1}^n to {0, 1}$ be a boolean function, and let $f_land (x, y) = f(x land y)$ denote the AND-function of $f$, where $x land y$ denotes bit-wise AND. We study the deterministic communication complexity of $f_land$ and show that, up to a $log n$ factor, it is bounded by a polynomial in the logarithm of the real rank of the communication matrix of $f_land$. This comes within a $log n$ factor of establishing the log-rank conjecturefor AND-functions with no assumptions on $f$. Our result stands in contrast with previous results on special cases of the log-rank conjecture, which needed significant restrictions on $f$ such as monotonicity or low $mathbb{F}_2$-degree. Our techniques can also be used to prove (within a $log n$ factor) a lifting theorem for AND-functions, stating that the deterministic communication complexity of $f_land$ is polynomially-related to the AND-decision tree complexity of $f$. The results rely on a new structural result regarding boolean functions $f:{0, 1}^n to {0, 1}$ with a sparse polynomial representation, which may be of independent interest. We show that if the polynomial computing $f$ has few monomials then the set system of the monomials has a small hitting set, of size poly-logarithmic in its sparsity. We also establish extensions of this result to multi-linear polynomials $f:{0,1}^n to mathbb{R}$ with a larger range.
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