Fourier uniformity of bounded multiplicative functions in short intervals on average

Let $\lambda$ denote the Liouville function. We show that as $X \rightarrow \infty$,$$ \int_{X}^{2X} \sup_{\alpha} \left | \sum_{x < n \leq x + H} \lambda(n) e(-\alpha n) \right | dx = o ( X H) $$forall$H \geq X^{\theta}$with$\theta > 0$ fixed but arbitrarily small. Previously, this was only known for $\theta >5\mathrm{/}8$. For smaller values of $\theta$ 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 $\lambda (n)\mathrm{\Lambda }(n+h)\mathrm{\Lambda }(n+2h)$ over the ranges $h<X^{\theta }$ and $n<X$, and where $\mathrm{\Lambda }$ is the von Mangoldt function.