Fourier uniformity of bounded multiplicative functions in short intervals on average - Kaisa Matomäki; Maksym Radziwiłł; Terence Tao (Springer New York LLC, 2022)
Let   λ  denote the Liouville function. We show that as   X→∞ ,
∫2XXsupα∣∣∣∣∑x
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+2h)  over the ranges   h