We establish a precise relationship between spherical harmonics and Fourier basis functions over a hypercube randomly embedded in the sphere. In particular, we give a bound on the expected Boolean noise sensitivity of a randomly rotated function in t
erms of its spherical sensitivity, which we define according to its evolution under the spherical heat equation. As an application, we prove an average case of the Gotsman-Linial conjecture, bounding the sensitivity of polynomial threshold functions subjected to a random rotation.
In analogy with epsilon-biased sets over Z_2^n, we construct explicit epsilon-biased sets over nonabelian finite groups G. That is, we find sets S subset G such that | Exp_{x in S} rho(x)| <= epsilon for any nontrivial irreducible representation rho.
Equivalently, such sets make Gs Cayley graph an expander with eigenvalue |lambda| <= epsilon. The Alon-Roichman theorem shows that random sets of size O(log |G| / epsilon^2) suffice. For groups of the form G = G_1 x ... x G_n, our construction has size poly(max_i |G_i|, n, epsilon^{-1}), and we show that a set S subset G^n considered by Meka and Zuckerman that fools read-once branching programs over G is also epsilon-biased in this sense. For solvable groups whose abelian quotients have constant exponent, we obtain epsilon-biased sets of size (log |G|)^{1+o(1)} poly(epsilon^{-1}). Our techniques include derandomized squaring (in both the matrix product and tensor product senses) and a Chernoff-like bound on the expected norm of the product of independently random operators that may be of independent interest.
Subsets of F_2^n that are eps-biased, meaning that the parity of any set of bits is even or odd with probability eps close to 1/2, are powerful tools for derandomization. A simple randomized construction shows that such sets exist of size O(n/eps^2),
and known deterministic constructions achieve sets of size O(n/eps^3), O(n^2/eps^2), and O((n/eps^2)^{5/4}). Rather than derandomizing these sets completely in exchange for making them larger, we attempt a partial derandomization while keeping them small, constructing sets of size O(n/eps^2) with as few random bits as possible. The naive randomized construction requires O(n^2/eps^2) random bits. We give two constructions. The first uses Nisans space-bounded pseudorandom generator to partly derandomize a folklore probabilistic construction of an error-correcting code, and requires O(n log (1/eps)) bits. Our second construction requires O(n log (n/eps)) bits, but is more elementary; it adds randomness to a Legendre symbol construction on Alon, Goldreich, H{aa}stad, and Peralta, and uses Weil sums to bound high moments of the bias.
Changs lemma is a useful tool in additive combinatorics and the analysis of Boolean functions. Here we give an elementary proof using entropy. The constant we obtain is tight, and we give a slight improvement in the case where the variables are highly biased.
