Boolean Functions with Minimal Spectral Sensitivity
Krišjānis Prūsis, Jevgēnijs Vihrovs
TL;DR
The paper investigates the minimal possible spectral sensitivity λ(f) for total non-degenerate Boolean functions that depend on all n variables, establishing that λ(f) = Θ(√log n) is asymptotically optimal. It presents two principal constructions: a Hamming-address function achieving λ ≈ √log n, and desensitized variants that initially reach larger constants but can be composed to preserve the optimal spectral bound while tuning the split between s0 and s1. A key contribution is showing an optimal tradeoff between 0- and 1-sensitivity under the minimal spectral sensitivity, yielding a tunable range s0 ≈ c log n and s1 ≈ (1−c) log n for any c ∈ [0,1], and implying the existence of a function with near-minimal total sensitivity s(f) ≈ (1/2) log n. These results sharpen the understanding of the Sensitivity Conjecture by linking spectral sensitivity to fundamental limits on sensitivity, degree, and certificate complexity, and provide explicit function families achieving these limits.
Abstract
We show examples of total Boolean functions that depend on $n$ variables and have spectral sensitivity $Θ(\sqrt{\log n})$, which is asymptotically minimal. Our main new function combines the Hamming code with the Boolean address function and has $λ(f) = \sqrt{(1+o(1)) \log_2 n}$, which is optimal even up to a constant factor. By combining this function with itself in a specific way, we also obtain a family of functions with $\text{s}_0(f) = (c+o(1)) \log_2 n$ and $\text{s}_0(f) = (1-c+o(1)) \log_2 n$ for any $c \in [0,1]$. This is an optimal tradeoff for Boolean functions with low sensitivity, as the lower bound on sensitivity by Simon generalizes to \[\text{s}_0(f)+\text{s}_1(f)\geq\log_2 n - \log_2 \log_2 n + 2.\] As a corollary, this gives a new example of a function with minimal possible sensitivity (up to a constant factor), $\text{s}(f) = (\frac{1}{2}+o(1)) \log_2 n$.