A forced coordinate with nearly maximal erasure entropy

The erasure entropy $E_x(f)$ of a Boolean function at a coordinate $x$ is the conditional entropy of the event $x\in U$ given $U\setminus\{x\}$, where $U$ is drawn from the spectral distribution, and it never exceeds the binary entropy of the influence $\mathrm{Inf}_x(f)$. We study an explicit monotone function on $2k+1$ variables in which one coordinate $x$ is the only variable lying both in a shortest minterm and in a shortest maxterm. It is obtained from a majority function with one hidden coordinate by replacing half of the majority's inputs with the OR of two fresh variables and the other half with the AND of two. We prove that $E_x/\mathrm{Inf}_x=(1+o(1))\log_2(1/\mathrm{Inf}_x)=(1+o(1))(k-1)\log_2\frac43$, so the ratio at the forced coordinate grows linearly in $k$ and is asymptotic to its largest possible value. The proof writes the relevant biased Fourier coefficients of the majority as integrals over the unit circle, on which the integrand has an explicit modulus with a single pair of peaks, and uses that the phase advances by $\arccos\frac13$, an irrational multiple of $\pi$, from one class of coefficients to the next.