Boolean Functions with Minimal Spectral Sensitivity

TL;DR

This paper constructs Boolean functions with minimal spectral sensitivity close to the theoretical lower bounds, using a novel combination of Hamming code and address functions, advancing understanding of sensitivity tradeoffs.

Contribution

It introduces a new Boolean function with spectral sensitivity matching the asymptotic minimal, and explores optimal sensitivity tradeoffs for low-sensitivity functions.

Findings

Constructed functions with spectral sensitivity ( ext{spectral sensitivity} o ext{minimal})

Established optimal tradeoffs between sensitivity parameters _0(f) and _1(f)

Provided examples of functions with near-minimal total sensitivity (f)

Abstract

We show examples of total Boolean functions that depend on $n$ variables and have spectral sensitivity $\Theta(\sqrt{\log n})$, which is asymptotically minimal. Our main new function combines the Hamming code with the Boolean address function and has $\lambda(f) = \sqrt{(1+o(1)) \log_2 n}$, 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), $\text{s}(f) = (\frac{1}{2}+o(1)) \log_2 n$.