Fourier uniformity of bounded multiplicative functions in short intervals on average (vol 220, pg 1, 2020)

Let λ denote the Liouville function. We show that as X→ ∞, ∫X2Xsupα|∑x<n≤x+Hλ(n)e(-αn)|dx=o(XH)for all H≥ Xθ 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+ 2 h) over the ranges h< Xθ and n< X, and where Λ is the von Mangoldt function. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.