Analyzing Boolean functions on the biased hypercube via higher-dimensional agreement tests

We propose a new paradigm for studying the structure of Boolean functions on the biased Boolean hypercube, i.e. when the measure is µ_p and p is potentially very small, e.g. as small as O(1/n). Our paradigm is based on the following simple fact: the p-biased hypercube is expressible as a convex combination of many small-dimensional copies of the uniform hypercube. To uncover structure for µ_p, we invoke known structure theorems for µ1/2, obtaining a structured approximation for each copy separately. We then sew these approximations together using a novel “agreement theorem”. This strategy allows us to lift structure theorems from µ1/₂ to µ_p. We provide two applications of this paradigm: • Our main application is a structure theorem for functions that are nearly low degree in the Fourier sense. The structure we uncover in the biased hypercube is not at all the same as for the uniform hypercube, despite using the structure theorem for the uniform hypercube as a black box. Rather, new phenomena emerge: whereas nearly low degree functions on the uniform hypercube are close to juntas, when p becomes small, non-juntas arise as well. For example, the function max(y₁, · · ·, y_ε/p) (where y_i ∈ {0, 1}) is nearly degree 1 despite not being close to any junta. • A second (technically simpler) application is a test for being low degree in the GF(2) sense, in the setting of the biased hypercube. In both cases, we use as a black box the corresponding result for p = 1/2. In the first case, it is the junta theorem of Kindler and Safra, and in the second case, the low degree testing theorem of Alon et al. [IEEE Trans. Inform. Theory, 2005] and Bhattacharyya et al. [Proc. 51st FOCS, 2010]. A key component of our proof is a new local-to-global agreement theorem for higher dimensions, which extends the work of Dinur and Steurer [Proc. 29th CCC, 2014]. Whereas their result sews together vectors, our agreement theorem sews together labeled graphs and hypergraphs. The proof of our agreement theorem uses a novel pruning lemma for hypergraphs, which may be of independent interest. The pruning lemma trims a given hypergraph so that the number of hyperedges in a random induced subhypergraph has roughly a Poisson distribution, while maintaining the expected number of hyperedges.