Talagrand’s “Convolution with a Biased Coin” Conjecture

Prove that for every function f: {−1,1}^n → R≥0 with E[f] = 1 and every 0 < ρ < 1, the noise operator satisfies Pr[Tρ f ≥ t] = o(1/t) as t → ∞.

Background

Talagrand suggested a stronger bound O(1/(t√log t)) and offered a monetary prize for a proof. The conjecture remains challenging even in Gaussian space.

In the Gaussian setting, known results establish O(1/(t√log t)) for dimension 1 and O((log log t)/(t√log t)) in fixed constant dimensions, but the general case is still unresolved.