Uniform Fourier decay for word measures on U_n

Establish that for every non-trivial word w in the free group F_r there exists a constant ε(w) > 0 such that, for all integers n ≥ 1 and every irreducible character ρ of the unitary group U_n, the Fourier coefficient of the word measure satisfies |E(ρ(w_{U_n}(G_1, …, G_r)))| ≤ ρ(1)^{1−ε(w)}, where G_1, …, G_r are independent Haar-distributed random elements of U_n and ρ(1) is the degree of ρ.

Background

The paper studies word maps and associated word measures on unitary and special unitary groups, focusing on small ball estimates and decay of Fourier coefficients. A central quantitative objective is to bound the Fourier coefficients of word measures uniformly across the family {U_n}.

The authors provide strong evidence toward a uniform Fourier decay statement by proving exponential bounds for high-degree characters (Theorem B), establishing bounds for low-degree characters (Theorem C), and resolving the case of power words (Theorem D). They also show bounded-density convolution results answering a separate conjecture about mixing, but the full uniform bound across all irreducible characters and all n remains unresolved in general.