核心概念
There exists a complex function f from {-1, 1}^n to the unit sphere in C with influence bounded by 1 and entropy of |f̂|^2 larger than (1/2) log n.
要約
The content presents an example of a complex function f from the Cantor group {-1, 1}^n to the complex plane C, with modulus 1, that has bounded influence but high entropy.
The key highlights are:
The function F is defined as (1 + 1/n)^(-n/2) * Π(1 + iε_j/√n), where ε_j are the projections onto the j-th coordinate.
The influence I(F) of this function is calculated to be n/(n+1), which is bounded by 1.
The entropy H(|F̂|^2) of the Fourier coefficients of F is shown to be greater than (n log n)/(n+1), which is larger than (1/2) log n for large n.
This provides a counterexample to the analogous version of the Friedgut-Kalai entropy-influence conjecture for complex functions of modulus one, as posed by Gady Kozma.
The author notes that this simple example was likely known to experts, but not published before.
統計
I(F) = n/(n+1)
H(|F̂|^2) > (n log n)/(n+1)