We study higher uniformity properties of the von Mangoldt function $\Lambda $ , the Möbius function $\mu $ , and the divisor functions $d_{k}$ on short intervals $(x,x+H]$ for almost all $x \in [X, 2X]$ . Let $\Lambda ^{\sharp }$ and $d_{k}^{\sharp }$ be suitable approximants of $\Lambda $ and $d_{k}$ , $G/\Gamma $ a filtered nilmanifold, and $F \colon G/\Gamma \to \mathbb{C}$ a Lipschitz function. Then our results imply for instance that when $X^{1/3+\varepsilon } \leq H \leq X$ we have, for almost all $x \in [X, 2X]$ , \( \sup _{g \in {\operatorname{Poly}}(\mathbb{Z}\to G)} \left | \sum _{x < n \leq x+H} (\Lambda (n)-\Lambda ^{\sharp }(n)) \overline{F}(g(n) \Gamma ) \right | \ll H\log ^{-A} X \) for any fixed $A>0$ , and that when $X^{\varepsilon } \leq H \leq X$ we have, for almost all $x \in [X, 2X]$ , \( \sup _{g \in {\operatorname{Poly}}(\mathbb{Z}\to G)} \left | \sum _{x < n \leq x+H} (d_{k}(n)-d_{k}^{\sharp }(n)) \overline{F}(g(n)\Gamma ) \right | = o(H \log ^{k-1} X). \) As a consequence, we show that the short interval Gowers norms $\|\Lambda -\Lambda ^{\sharp }\|_{U^{s}(X,X+H]}$ and $\|d_{k}-d_{k}^{\sharp }\|_{U^{s}(X,X+H]}$ are also asymptotically small for any fixed $s$ in the same ranges of $H$ . This in turn allows us to establish the Hardy–Littlewood conjecture and the divisor correlation conjecture with a short average over one variable. Our main new ingredients are type $\mathit{II}$ estimates obtained by developing a “contagion lemma” for nilsequences and then using this to “scale up” an approximate functional equation for the nilsequence to a larger scale. This extends an approach developed by Walsh for Fourier uniformity.
