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Abstract We study higher uniformity properties of the von Mangoldt function$$\Lambda $$ $\Lambda$ , the Möbius function$$\mu $$ $\mu$ , and the divisor functions$$d_{k}$$ $d_k$ on short intervals$$(x,x+H]$$ $(x,x+H]$ for almost all$$x \in [X, 2X]$$ $x\in [X,2X]$ . Let$$\Lambda ^{\sharp }$$ ${\Lambda }^{♯}$ and$$d_{k}^{\sharp }$$ $d_k^{♯}$ be suitable approximants of$$\Lambda $$ $\Lambda$ and$$d_{k}$$ $d_k$ ,$$G/\Gamma $$ $G/\Gamma$ a filtered nilmanifold, and$$F \colon G/\Gamma \to \mathbb{C}$$ $F:G/\Gamma \to C$ a Lipschitz function. Then our results imply for instance that when$$X^{1/3+\varepsilon } \leq H \leq X$$ $X^{1/3+\epsilon }\le H\le X$ we have, for almost all$$x \in [X, 2X]$$ $x\in [X,2X]$ ,$$ \sup _{g \in {\operatorname{Poly}}(\mathbb{Z}\to G)} \left | \sum _{x < n \leq x+H} (\Lambda (n)-\Lambda ^{\sharp }(n)) \overline{F}(g(n) \Gamma ) \right | \ll H\log ^{-A} X $$ $\underset{g\in Poly(Z\to G)}{sup}\left|\sum _{x<n\le x+H}\right(\Lambda \left(n\right)-{\Lambda }^{♯}\left(n\right)\left)\bar{F}\right(g\left(n\right)\Gamma \left)\right|\ll H{log}^{-A}X$ for any fixed$$A>0$$ $A>0$ , and that when$$X^{\varepsilon } \leq H \leq X$$ $X^{\epsilon }\le H\le X$ we have, for almost all$$x \in [X, 2X]$$ $x\in [X,2X]$ ,$$ \sup _{g \in {\operatorname{Poly}}(\mathbb{Z}\to G)} \left | \sum _{x < n \leq x+H} (d_{k}(n)-d_{k}^{\sharp }(n)) \overline{F}(g(n)\Gamma ) \right | = o(H \log ^{k-1} X). $$ $\underset{g\in Poly(Z\to G)}{sup}\left|\sum _{x<n\le x+H}\right(d_k\left(n\right)-d_k^{♯}\left(n\right)\left)\bar{F}\right(g\left(n\right)\Gamma \left)\right|=o\left(H{log}^{k-1}X\right).$ As a consequence, we show that the short interval Gowers norms$$\|\Lambda -\Lambda ^{\sharp }\|_{U^{s}(X,X+H]}$$ ${\parallel \Lambda -{\Lambda }^{♯}\parallel }_{U^s(X,X+H]}$ and$$\|d_{k}-d_{k}^{\sharp }\|_{U^{s}(X,X+H]}$$ ${\parallel d_k-d_k^{♯}\parallel }_{U^s(X,X+H]}$ are also asymptotically small for any fixed$$s$$ $s$ in the same ranges of$$H$$ $H$ . 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$$\mathit{II}$$ $\mathrm{II}$ 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. 

In the 1960s, Berger famously showed that translational tilings of\mathbb{Z}^{2}with multiple tiles are algorithmically undecidable. Recently, Bhattacharya proved the decidability oftranslational monotilings(tilings by translations of a single tile) in\mathbb{Z}^{2}. The decidability of translational monotilings in higher dimensions remained unsolved. In this paper, by combining our recently developed techniques with ideas introduced by Aanderaa–Lewis, we finally settle this problem, achieving the undecidability of translational monotilings of (periodic subsets of) virtually\mathbb{Z}^{2}spaces, namely, spaces of the form\mathbb{Z}^{2}\times G_{0}, whereG_{0}is a finite Abelian group. This also implies the undecidability of translational monotilings in\mathbb{Z}^{d},d\geq 3. 

Abstract The entropic doubling of a random variable taking values in an abelian group is a variant of the notion of the doubling constant of a finite subset of , but it enjoys somewhat better properties; for instance, it contracts upon applying a homomorphism. In this paper we develop further the theory of entropic doubling and give various applications, including: (1) A new proof of a result of Pálvölgyi and Zhelezov on the “skew dimension” of subsets of with small doubling; (2) A new proof, and an improvement, of a result of the second author on the dimension of subsets of with small doubling; (3) A proof that the Polynomial Freiman–Ruzsa conjecture over implies the (weak) Polynomial Freiman–Ruzsa conjecture over . 

Abstract We give an improved lower bound for the average of the Erdős–Hooley function , namely for all and any fixed , where is an exponent previously appearing in work of Green and the first two authors. This improves on a previous lower bound of of Hall and Tenenbaum, and can be compared to the recent upper bound of of the second and third authors. 

Let $\mathrm{\Gamma <\#comment/>}$ be a countable abelian group. An (abstract) $\mathrm{\Gamma <\#comment/>}$ -system $\mathrm{X}$ - that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of $\mathrm{\Gamma <\#comment/>}$ - is said to be aConze–Lesigne systemif it is equal to its second Host–Kra–Ziegler factor ${\mathrm{Z}}^2\left(\mathrm{X}\right)$ . The main result of this paper is a structural description of such Conze–Lesigne systems for arbitrary countable abelian $\mathrm{\Gamma <\#comment/>}$ , namely that they are the inverse limit of translational systems $G_n/{\mathrm{\Lambda <\#comment/>}}_n$ arising from locally compact nilpotent groups $G_n$ of nilpotency class $2$ , quotiented by a lattice ${\mathrm{\Lambda <\#comment/>}}_n$ . Results of this type were previously known when $\mathrm{\Gamma <\#comment/>}$ was finitely generated, or the product of cyclic groups of prime order. In a companion paper, two of us will apply this structure theorem to obtain an inverse theorem for the Gowers $U^3\left(G\right)$ norm for arbitrary finite abelian groups $G$ .