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Kindler-Safra Theorem on the p-biased hypercube via agreement theorems Harsha, Prahladh


Nisan and Szegedy showed that low degree Boolean functions are juntas, namely, they depend only on a constant number of their variables. Kindler and Safra showed a robust version of the above: low degree functions which are almost Boolean are close to juntas. We study the same question on the p-biased hypercube, for very small p. The p-biased hypercube is a product probability space in which each coordinate is 1 with probability p and 0 otherwise. In this space most of the measure is on n-bit strings whose Hamming weight about pn << n. It turns out that here new phenomena emerge. For example, the function x_1 + ... + x_n=p (where x_i \in {0,1}) is close to Boolean but not close to a junta. We characterize low degree functions that are almost Boolean and show that they are close to a new class of functions which we call sparse juntas. An interesting aspect of our proof is a new proof paradigm that relies on a local to global agreement theorem. We cover the p-biased hypercube by many smaller dimensional copies of the uniform hypercube, and approximate our function locally via the standard Kindler-Safra theorem for constant p. We then stitch the local approximations together into one global function that is a sparse junta. The stitching is made feasible via a new local-to-global agreement theorem, which is an extension of the classical direct product results to larger dimensions. Time permitting, I'll show another application of this paradigm: extending the classical AKKLR low-degree tests to the p-biased hypercube. Based on joint work with Irit Dinur and Yuval Filmus.

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