Fourier uniformity of bounded multiplicative functions in short intervals on average

Abstract

Let λ denote the Liouville function. We show that as X→∞, ∫^(2X)X supα∣∑x< n ≤ x+H λ(n)e(−αn)∣dx = o(XH) for all H ≥ X^θ with θ > 0 fixed but arbitrarily small. Previously, this was only known for θ > 5/8. For smaller values of θ this is the first “non-trivial” case of local Fourier uniformity on average at this scale. We also obtain the analogous statement for (non-pretentious) 1-bounded multiplicative functions. We illustrate the strength of the result by obtaining cancellations in the sum of λ(n)Λ(n+h)Λ(n+2h) over the ranges h < X^θ and n < X, and where Λ is the von Mangoldt function.