Higher uniformity of arithmetic functions in short intervals II. Almost all intervals

Abstract

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 ${\Lambda }^{♯}$ and $d_k^{♯}$ be suitable approximants of $\Lambda$ and $d_k$, $G/\Gamma$ a filtered nilmanifold, and $F:G/\Gamma \to \mathbb{C}$ a Lipschitz function. Then our results imply for instance that when $X^{1/3+\epsilon }\le H\le X$ we have, for almost all $x\in [X,2X]$, \[ \sup_{g \in \text{Poly}(\mathbb{Z} \to G)} \left| \sum_{x < n \leq x+H} (Λ(n)-Λ^\sharp(n)) \overline{F}(g(n)Γ) \right| \ll H\log^{-A} X \] for any fixed $A>0$, and that when $X^{\epsilon }\le H\le X$ we have, for almost all $x\in [X,2X]$, \[ \sup_{g \in \text{Poly}(\mathbb{Z} \to G)} \left| \sum_{x < n \leq x+H} (d_k(n)-d_k^\sharp(n)) \overline{F}(g(n)Γ) \right| = o(H \log^{k-1} X). \] As a consequence, we show that the short interval Gowers norms $\Vert \Lambda -{\Lambda }^{♯}{\Vert }_{U^s(X,X+H]}$ and $\Vert d_k-d_k^{♯}{\Vert }_{U^s(X,X+H]}$ 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 $II$ 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.

105 pages; referee comments incorporated; to appear in Invent. Math