59 research outputs found
Boolean constant degree functions on the slice are juntas
We show that a Boolean degree function on the slice is a junta, assuming
that are large enough. This generalizes a classical result of Nisan and
Szegedy on the hypercube. Moreover, we show that the maximum number of
coordinates that a Boolean degree function can depend on is the same on the
slice and the hypercube.Comment: 10 page
Invariance principle on the slice
We prove an invariance principle for functions on a slice of the Boolean
cube, which is the set of all vectors {0,1}^n with Hamming weight k. Our
invariance principle shows that a low-degree, low-influence function has
similar distributions on the slice, on the entire Boolean cube, and on Gaussian
space.
Our proof relies on a combination of ideas from analysis and probability,
algebra and combinatorics.
Our result imply a version of majority is stablest for functions on the
slice, a version of Bourgain's tail bound, and a version of the Kindler-Safra
theorem. As a corollary of the Kindler-Safra theorem, we prove a stability
result of Wilson's theorem for t-intersecting families of sets, improving on a
result of Friedgut.Comment: 36 page
Harmonicity and Invariance on Slices of the Boolean Cube
In a recent work with Kindler and Wimmer we proved an invariance principle for the slice for low-influence, low-degree functions. Here we provide an alternative proof for general low-degree functions, with no constraints on the influences. We show that any real-valued function on the slice, whose degree when written as a harmonic multi-linear polynomial is o(sqrt(n)), has approximately the same distribution under the slice and cube measure.
Our proof is based on a novel decomposition of random increasing paths in the cube in terms of martingales and reverse martingales. While such decompositions have been used in the past for stationary reversible Markov chains, ours decomposition is applied in a non-reversible non-stationary setup. We also provide simple proofs for some known and some new properties of harmonic functions which are crucial for the proof.
Finally, we provide independent simple proofs for the known facts that 1) one cannot distinguish between the slice and the cube based on functions of little of of n coordinates and 2) Boolean symmetric functions on the cube cannot be approximated under the uniform measure by functions whose sum of influences is o(sqrt(n))
Boolean Function Analysis on High-Dimensional Expanders
We initiate the study of Boolean function analysis on high-dimensional expanders. We describe an analog of the Fourier expansion and of the Fourier levels on simplicial complexes, and generalize the FKN theorem to high-dimensional expanders.
Our results demonstrate that a high-dimensional expanding complex X can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing |X(k)|=O(n) points in comparison to binom{n}{k+1} points in the (k+1)-slice (which consists of all n-bit strings with exactly k+1 ones)
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