Fourier uniformity of bounded multiplicative functions in short intervals on average
Let \lambda l denote the Liouville function. We show that as $X \rightarrow \infty X \begin{aligned} \int _{X}{2X} \sup _{\alpha \left| \sum _{x n \le x H} \lambda (n) e( \alpha n) \right| dx o X H) \end{aligned} X 2 X sup a x n x H l n e a n d x o X H for all $H \ge X^\theta H X th with \theta 0$ th 0 fixed but arbitrarily small. Previously, this was only known for \theta 5/8$ th 5 8 For smaller values of \theta th 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 \lambda (n) \Lambda (n h) \Lambda (n 2h) l n L n h L n 2 h over the ranges $h X^\theta h X th and $n X$ n X and where \Lambda L is the von Mangoldt function.
Inventiones mathematicae