Stability under scaling in the local phases of multiplicative functions

Abstract

We introduce a strategy to tackle some known obstructions of current approaches to the Fourier uniformity conjecture. Assuming GRH, we then show the conjecture holds for intervals of length at least (log X)^{psi(X)}, with psi(X) -> infty an arbitrarily slowly growing function of X. We expect the methods should adapt to nilsequences, thus also showing that the Generalised Riemann Hypothesis implies close to exponential growth in the sign patterns of the Liouville function.