Sign Changes of the Error Term in the Piltz Divisor ProblemBaluyot, Siegfred; Castillo, Cruz
doi: 10.1093/imrn/rnae189pmid: N/A
We study the function $\Delta _{k}(x):=\sum _{n\leq x} d_{k}(n) - \mbox{Res}_{s=1} ( \zeta ^{k}(s) x^{s}/s )$, where $k\geq 3$ is an integer, $d_{k}(n)$ is the $k$-fold divisor function, and $\zeta (s)$ is the Riemann zeta-function. For a large parameter $X$, we show that if the Lindelöf hypothesis (LH) is true, then there exist at least $X^{\frac{1}{k(k-1)}-\varepsilon }$ disjoint subintervals of $[X,2X]$, each of length $X^{1-\frac{1}{k}-\varepsilon }$, such that $|\Delta _{k}(x)|\gg x^{\frac{1}{2}-\frac{1}{2k}}$ for all $x$ in the subinterval. In particular, $\Delta _{k}(x)$ does not change sign in any of these subintervals. If the Riemann hypothesis (RH) is true, then we can improve the length of the subintervals to $\gg X^{1-\frac{1}{k}} (\log X)^{-k^{2}-2}$. These results may be viewed as higher-degree analogues of theorems of Heath-Brown and Tsang, who studied the case $k=2$, and Cao, Tanigawa, and Zhai, who studied the case $k=3$. The first main ingredient of our proofs is a bound for the second moment of $\Delta _{k}(x+h)-\Delta _{k}(x)$. We prove this bound using a method of Selberg and a general lemma due to Saffari and Vaughan. The second main ingredient is a bound for the fourth moment of $\Delta _{k}(x)$, which we obtain by combining a method of Tsang with a technique of Lester.
Uniqueness and Non-Uniqueness Results for Spacetime ExtensionsSbierski, Jan
doi: 10.1093/imrn/rnae194pmid: N/A
Given a function $f: A \to{\mathbb{R}}^{n}$ of a certain regularity defined on some open subset $A \subseteq{\mathbb{R}}^{m}$, it is a classical problem of analysis to investigate whether the function can be extended to all of ${\mathbb{R}}^{m}$ in a certain regularity class. If an extension exists and is continuous, then certainly it is uniquely determined on the closure of $A$. A similar problem arises in general relativity for Lorentzian manifolds instead of functions on ${\mathbb{R}}^{m}$. It is well-known, however, that even if the extension of a Lorentzian manifold $(M,g)$ is analytic, various choices are in general possible at the boundary. This paper establishes a uniqueness condition for extensions of globally hyperbolic Lorentzian manifolds $(M,g)$ with a focus on low regularities: any two extensions that are anchored by an inextendible causal curve $\gamma : [-1,0) \to M$ in the sense that $\gamma $ has limit points in both extensions must agree locally around those limit points on the boundary as long as the extensions are at least locally Lipschitz continuous. We also show that this is sharp: anchored extensions that are only Hölder continuous do in general not enjoy this local uniqueness result.
On the Fourier Coefficients of Powers of a Finite Blaschke ProductBorichev, Alexander; Fouchet, Karine; Zarouf, Rachid
doi: 10.1093/imrn/rnae199pmid: N/A
Given a finite Blaschke product $B$ we prove asymptotically sharp estimates on the $\ell ^{\infty }$-norm of the sequence of the Fourier coefficients of $B^{n}$ as $n$ tends to $\infty $. This norm decays as $n^{-1/N}$ for some $N\ge 3$. Furthermore, for every $N\ge 3$, we produce explicitly a finite Blaschke product $B$ with decay $n^{-1/N}$. As an application we construct a sequence of $n\times n$ invertible matrices $T$ with arbitrary spectrum in the unit disk and such that the quantity $|\det{T}|\cdot \|T^{-1}\|\cdot \|T\|^{1-n}$ grows as a power of $n$. This is motivated by Schäffer’s question on norms of inverses.
Gravitational Instantons, Weyl Curvature, and Conformally Kähler GeometryBiquard, Olivier; Gauduchon, Paul; LeBrun, Claude
doi: 10.1093/imrn/rnae200pmid: N/A
In a previous paper [7], the first two authors classified complete Ricci-flat ALF Riemannian 4-manifolds that are toric and Hermitian, but non-Kähler. In this article, we consider general Ricci-flat metrics on these spaces that are close to a given such gravitational instanton with respect to a norm that imposes reasonable fall-off conditions at infinity. We prove that any such Ricci-flat perturbation is necessarily Hermitian and carries a bounded, non-trivial Killing vector field. With mild additional hypotheses, we are then able to show that the new Ricci-flat metric must actually be one of the known gravitational instantons classified in [7].
Hopf Monoids of Ordered Simplicial ComplexesCastillo, Federico; Martin, Jeremy L; Samper, José A
doi: 10.1093/imrn/rnae201pmid: N/A
We study Hopf classes: families of pure ordered simplicial complexes that give rise to Hopf monoids under join and deletion/contraction. The prototypical Hopf class is the family of ordered matroids. The idea of a Hopf class leads to a systematic study of simplicial complexes related to matroids, including shifted complexes and broken-circuit complexes. We compute the Hopf antipodes in two cases: facet-initial complexes (which generalize shifted complexes) and unbounded ordered matroids. The latter calculation uses the topological method of Aguiar and Ardila, complicated by certain auxiliary simplicial complexes that we call Scrope complexes, whose Euler characteristics control the coefficients of the antipode. The resulting antipode formula is multiplicity-free and cancellation-free.
On the v-Picard Group of Stein SpacesErtl, Veronika; Gilles, Sally; Nizioł, Wiesława
doi: 10.1093/imrn/rnae185pmid: N/A
We study the image of the Hodge–Tate logarithm map (in any cohomological degree), defined by Heuer, in the case of smooth Stein varieties. Heuer, motivated by the computations for the affine space of any dimension, raised the question whether this image is always equal to the group of closed differential forms. We show that it indeed always contains such forms but the quotient can be non-trivial: it contains a slightly mysterious $\mathbf{Z}_{p}$-module that maps, via the Bloch–Kato exponential map, to integral classes in the pro-étale cohomology. This quotient is already non-trivial for open unit disks of dimension strictly greater than $1$.
The HHMP Decomposition of the Permutohedron and Degenerations of Torus Orbits in Flag VarietiesLian, Carl
doi: 10.1093/imrn/rnae204pmid: N/A
Let $Z\subset \operatorname{Fl}(n)$ be the closure of a generic torus orbit in the full flag variety. Anderson–Tymoczko express the cohomology class of $Z$ as a sum of classes of Richardson varieties. Harada–Horiguchi–Masuda–Park give a decomposition of the permutohedron, the moment map image of $Z$, into subpolytopes corresponding to the summands of the Anderson–Tymoczko formula. We construct an explicit toric degeneration inside $\operatorname{Fl}(n)$ of $Z$ into Richardson varieties, whose moment map images coincide with the HHMP decomposition, thereby obtaining a new proof of the Anderson–Tymoczko formula.