Higher uniformity of bounded multiplicative functions in short intervals on average

Let $\lambda$ denote the Liouville function. We show that, as $X \rightarrow \infty$, $$\int_{X}^{2X} \sup_{\substack{P(Y)\in \mathbb{R}[Y]\\ deg(P)\leq k}} \Big | \sum_{x \leq n x + H} \lambda(n) e(-P(n)) |\ dx = o ( X H)$$ for all fixed $k$ and $X^{\theta} H X$ with $0 < \theta 1$ but arbitrarily small. Previously this was only established $k 1$. obtain result a special case of corresponding statement (non-pretentious) $1$-bounded multiplicative functions that we prove. In fact, are able to replace polynomial phases $e(-P(n))$ by degree nilsequences $\overline{F}(g(n) \Gamma)$. By inverse theory Gowers norms implies higher order asymptotic uniformity \| \lambda \|_{U^{k+1}([x,x+H])}\ )$$ in same range $H$. present applications patterns various types sequence. Firstly, number sign function is superpolynomial, making progress on conjecture Sarnak about sequence having positive entropy. Secondly, cancellation averages over short progressions $(n+P_1(m),\ldots, n+P_k(m))$, which linear polynomials yields new averaged version Chowla's conjecture. fact prove our results wider $H\geq \exp((\log X)^{5/8+\varepsilon})$, thus strengthening also previous work Fourier