Local Fourier uniformity of higher divisor functions on average

Let τₖ be the k-fold divisor function. By constructing an approximant of τₖ, denoted as τₖ^*, which is a normalized truncation of the k-fold divisor function, we prove that when exp(Clog1/2X(loglog X)1/2)≤ H≤ X and $C>0$ is sufficiently large, the following estimate holds for almost all x∈[X,2X]: \[ |∑x<n≤ x+H(τ_k(n)-τ_k^*(n)) e(α_dn^d+⋯+α_1n)|=o(Hlogᵏ⁻¹X), \] where α₁, , αd∈ R are arbitrary frequencies.