Boolean Functions with Minimal Spectral Sensitivity

Abstract

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