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1. Khintchine-type recurrence for 3-point configurations

   https://doi.org/10.1017/fms.2022.97  

   Ackelsberg, Ethan; Bergelson, Vitaly; Shalom, Or (January 2022, Forum of Mathematics, Sigma)

   Abstract The goal of this paper is to generalise, refine and improve results on large intersections from [2, 8]. We show that if G is a countable discrete abelian group and $$\varphi , \psi : G \to G$$ are homomorphisms, such that at least two of the three subgroups $$\varphi (G)$$ , $$\psi (G)$$ and $$(\psi -\varphi )(G)$$ have finite index in G , then $$\{\varphi , \psi \}$$ has the large intersections property . That is, for any ergodic measure preserving system $$\textbf {X}=(X,\mathcal {X},\mu ,(T_g)_{g\in G})$$ , any $$A\in \mathcal {X}$$ and any $$\varepsilon>0$$ , the set $$ \begin{align*} \{g\in G : \mu(A\cap T_{\varphi(g)}^{-1} A \cap T_{\psi(g)}^{-1} A)>\mu(A)^3-\varepsilon\} \end{align*} $$ is syndetic (Theorem 1.11). Moreover, in the special case where $$\varphi (g)=ag$$ and $$\psi (g)=bg$$ for $$a,b\in \mathbb {Z}$$ , we show that we only need one of the groups $aG$ , $bG$ or $(b-a)G$ to be of finite index in G (Theorem 1.13), and we show that the property fails, in general, if all three groups are of infinite index (Theorem 1.14). One particularly interesting case is where $$G=(\mathbb {Q}_{>0},\cdot )$$ and $$\varphi (g)=g$$ , $$\psi (g)=g^2$$ , which leads to a multiplicative version of the Khintchine-type recurrence result in [8]. We also completely characterise the pairs of homomorphisms $$\varphi ,\psi $$ that have the large intersections property when $$G = {{\mathbb Z}}^2$$ . The proofs of our main results rely on analysis of the structure of the universal characteristic factor for the multiple ergodic averages $$ \begin{align*} \frac{1}{|\Phi_N|} \sum_{g\in \Phi_N}T_{\varphi(g)}f_1\cdot T_{\psi(g)} f_2. \end{align*} $$ In the case where G is finitely generated, the characteristic factor for such averages is the Kronecker factor . In this paper, we study actions of groups that are not necessarily finitely generated, showing, in particular, that, by passing to an extension of $$\textbf {X}$$ , one can describe the characteristic factor in terms of the Conze–Lesigne factor and the $$\sigma $$ -algebras of $$\varphi (G)$$ and $$\psi (G)$$ invariant functions (Theorem 4.10). 

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2. Boundary regularity and stability for spaces with Ricci bounded below

   https://doi.org/10.1007/s00222-021-01092-8  

   Bruè, Elia; Naber, Aaron; Semola, Daniele (May 2022, Inventiones mathematicae)

   Abstract This paper studies the structure and stability of boundaries in noncollapsed $${{\,\mathrm{RCD}\,}}(K,N)$$ RCD ( K , N ) spaces, that is, metric-measure spaces $$(X,{\mathsf {d}},{\mathscr {H}}^N)$$ ( X , d , H N ) with Ricci curvature bounded below. Our main structural result is that the boundary $$\partial X$$ ∂ X is homeomorphic to a manifold away from a set of codimension 2, and is $$N-1$$ N - 1 rectifiable. Along the way, we show effective measure bounds on the boundary and its tubular neighborhoods. These results are new even for Gromov–Hausdorff limits $$(M_i^N,{\mathsf {d}}_{g_i},p_i) \rightarrow (X,{\mathsf {d}},p)$$ ( M i N , d g i , p i ) → ( X , d , p ) of smooth manifolds with boundary, and require new techniques beyond those needed to prove the analogous statements for the regular set, in particular when it comes to the manifold structure of the boundary $$\partial X$$ ∂ X . The key local result is an $$\varepsilon $$ ε -regularity theorem, which tells us that if a ball $$B_{2}(p)\subset X$$ B 2 ( p ) ⊂ X is sufficiently close to a half space $$B_{2}(0)\subset {\mathbb {R}}^N_+$$ B 2 ( 0 ) ⊂ R + N in the Gromov–Hausdorff sense, then $$B_1(p)$$ B 1 ( p ) is biHölder to an open set of $${\mathbb {R}}^N_+$$ R + N . In particular, $$\partial X$$ ∂ X is itself homeomorphic to $$B_1(0^{N-1})$$ B 1 ( 0 N - 1 ) near $$B_1(p)$$ B 1 ( p ) . Further, the boundary $$\partial X$$ ∂ X is $$N-1$$ N - 1 rectifiable and the boundary measure "Equation missing" is Ahlfors regular on $$B_1(p)$$ B 1 ( p ) with volume close to the Euclidean volume. Our second collection of results involve the stability of the boundary with respect to noncollapsed mGH convergence $$X_i\rightarrow X$$ X i → X . Specifically, we show a boundary volume convergence which tells us that the $$N-1$$ N - 1 Hausdorff measures on the boundaries converge "Equation missing" to the limit Hausdorff measure on $$\partial X$$ ∂ X . We will see that a consequence of this is that if the $$X_i$$ X i are boundary free then so is X . 

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3. Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile

   https://doi.org/10.1007/s00454-022-00426-4  

   Greenfeld, Rachel; Tao, Terence (January 2023, Discrete & Computational Geometry)

   Abstract We construct an example of a group$$G = \mathbb {Z}^2 \times G_0$$ $G=Z^2\times G_0$for a finite abelian group $$G_0$$ $G_0$, a subsetEof $$G_0$$ $G_0$, and two finite subsets$$F_1,F_2$$ $F_1,F_2$of G, such that it is undecidable in ZFC whether$$\mathbb {Z}^2\times E$$ $Z^2\times E$can be tiled by translations of$$F_1,F_2$$ $F_1,F_2$. In particular, this implies that this tiling problem isaperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings ofEby the tiles$$F_1,F_2$$ $F_1,F_2$, but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in $$\mathbb {Z}^2$$ $Z^2$). A similar construction also applies for$$G=\mathbb {Z}^d$$ $G=Z^d$for sufficiently large d. If one allows the group$$G_0$$ $G_0$to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile F. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles. 

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4. On finite energy monopoles on $$C x \Sigma$$.

   Wang, Donghao (July 2019, ArXiv.org)

   Let $$X=\mathbb{C}\times\Sigma$$ be the product of the complex plane and a compact Riemann surface. We establish a classification theorem of solutions to the Seiberg-Witten equation on $$X$$ with finite analytic energy. The spin bundle $$S^+\to X$$ splits as $$L^+\oplus L^-$$. When $$2-2g\leq c_1(S^+)[\Sigma]<0$$, the moduli space is in bijection with the moduli space of pairs $$((L^+,\bar{\partial}), f)$$ where $$(L^+,\bar{\partial})$$ is a holomorphic structure on $L^+$ and $$f: \mathbb{C}\to H^0(\Sigma, L^+,\bar{\partial})$$ is a polynomial map. Moreover, the solution has analytic energy $$-4\pi^2d\cdot c_1(S^+)[\Sigma]$$ if $$f$$ has degree $$d$$. When $$c_1(S^+)=0$$, all solutions are reducible and the moduli space is the space of flat connections on $$\bigwedge^2 S^+$$. We also estimate the decay rate at infinity for these solutions. 

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5. Kato square root problem with unbounded leading coefficients

   https://doi.org/https://doi.org/10.1090/proc/14224  

   Escauriaza, L.; Hofmann, S. (October 2018, Proceedings of the American Mathematical Society)

   We prove the Kato conjecture for elliptic operators, $$L=-\nabla\cdot\left((\mathbf A+\mathbf D)\nabla\ \right)$$, with $$\mathbf A$$ a complex measurable bounded coercive matrix and $$\mathbf D$$ a measurable real-valued skew-symmetric matrix in $$\re^n$$ with entries in $$BMO(\re^n)$$;\, i.e., the domain of $$\sqrt{L}\,$$ is the Sobolev space $$\dot H^1(\re^n)$$, with the estimate $$\|\sqrt{L}\, f\|_2 \lesssim \| \nabla f\|_2\,.$$ 

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