Higher uniformity of arithmetic functions in short intervals II. Almost all intervals
Kaisa Matomäki, Maksym Radziwiłł, Xuancheng Shao, Terence Tao, Joni Teräväinen
TL;DR
The paper advances higher-order uniformity results for arithmetic functions on almost-all short intervals, establishing tight discorrelation bounds with nilsequences and deriving consequential Gowers norms estimates. Central to the approach are new contagion and scaling tools that propagate nilsequence structure across scales, combined with major-arc/inverse-theorem machinery to control Type II behavior. This yields asymptotic Hardy–Littlewood-type correlations with one averaging variable and sharp divisor-correlation results in short intervals, substantially strengthening the almost-all regime relative to previous all-interval results. The methods unify Heath–Brown decompositions, nilmanifold equidistribution, and a robust 컨tagion framework to push the analysis of μ, Λ, and d_k beyond prior barriers, with significant implications for prime tuples and divisor correlations in short intervals.
Abstract
We study higher uniformity properties of the von Mangoldt function $Λ$, the Möbius function $μ$, and the divisor functions $d_k$ on short intervals $(x,x+H]$ for almost all $x \in [X, 2X]$. Let $Λ^\sharp$ and $d_k^\sharp$ be suitable approximants of $Λ$ and $d_k$, $G/Γ$ a filtered nilmanifold, and $F\colon G/Γ\to \mathbb{C}$ a Lipschitz function. Then our results imply for instance that when $X^{1/3+\varepsilon} \leq H \leq X$ we have, for almost all $x \in [X, 2X]$, \[ \sup_{g \in \text{Poly}(\mathbb{Z} \to G)} \left| \sum_{x < n \leq x+H} (Λ(n)-Λ^\sharp(n)) \overline{F}(g(n)Γ) \right| \ll H\log^{-A} X \] for any fixed $A>0$, and that when $X^{\varepsilon} \leq H \leq X$ 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 $\|Λ-Λ^\sharp\|_{U^s(X,X+H]}$ 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 $II$ 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.