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1. VC-Dimension of Hyperplanes Over Finite Fields

   https://doi.org/10.1007/s00373-025-02909-6  

   Ascoli, Ruben; Betti, Livia; Cheigh, Justin; Iosevich, Alex; Jeong, Ryan; Liu, Xuyan; McDonald, Brian; Milgrim, Wyatt; Miller, Steven J; Romero_Acosta, Francisco; et al (April 2025, Graphs and Combinatorics)

   Abstract Let$$\mathbb {F}_q^d$$ $F_q^d$be thed-dimensional vector space over the finite field withqelements. For a subset$$E\subseteq \mathbb {F}_q^d$$ $E\subseteq F_q^d$and a fixed nonzero$$t\in \mathbb {F}_q$$ $t\in F_q$, let$$\mathcal {H}_t(E)=\{h_y: y\in E\}$$ $H_t\left(E\right)=\{h_y:y\in E\}$, where$$h_y:E\rightarrow \{0,1\}$$ $h_y:E\to \{0,1\}$is the indicator function of the set$$\{x\in E: x\cdot y=t\}$$ $\{x\in E:x·y=t\}$. Two of the authors, with Maxwell Sun, showed in the case$$d=3$$ $d=3$that if$$|E|\ge Cq^{\frac{11}{4}}$$ $\left|E\right|\ge C{q}^{\frac{11}{4}}$andqis sufficiently large, then the VC-dimension of$$\mathcal {H}_t(E)$$ $H_t\left(E\right)$is 3. In this paper, we generalize the result to arbitrary dimension by showing that the VC-dimension of$$\mathcal {H}_t(E)$$ $H_t\left(E\right)$isdwhenever$$E\subseteq \mathbb {F}_q^d$$ $E\subseteq F_q^d$with$$|E|\ge C_d q^{d-\frac{1}{d-1}}$$ $\left|E\right|\ge C_d{q}^{d-\frac{1}{d-1}}$. 

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2. Transversality of holomorphic maps into hyperquadrics

   https://doi.org/10.1007/s00208-025-03134-5  

   Huang, Xiaojun; Zhu, Weixia (June 2025, Mathematische Annalen)

   Abstract We study holomorphic mapsFfrom a smooth Levi non-degenerate real hypersurface$$ M_{\ell }\subset {\mathbb {C}}^n $$ $M_{\ell }\subset C^n$into a hyperquadric$$ {\mathbb {H}}_{\ell '}^N $$ $H_{{\ell }^{\prime }}^N$with signatures$$ \ell \le (n-1)/2 $$ $\ell \le (n-1)/2$and$$ \ell '\le (N-1)/2,$$ ${\ell }^{\prime }\le (N-1)/2,$respectively. Assuming that$$ N - n < n - 1,$$ $N-n<n-1,$we prove that if$$ \ell = \ell ',$$ $\ell ={\ell }^{\prime },$thenFis either CR transversal to$$ {\mathbb {H}}_{\ell }^N $$ $H_{\ell }^N$at every point of$$ M_{\ell },$$ $M_{\ell },$or it maps a neighborhood of$$ M_{\ell } $$ $M_{\ell }$in$$ {\mathbb {C}}^n $$ $C^n$into$$ {\mathbb {H}}_{\ell }^N.$$ $H_{\ell }^N.$Furthermore, in the case where$$ \ell ' > \ell ,$$ ${\ell }^{\prime }>\ell ,$we show that ifFis not CR transversal at$$0\in M_\ell ,$$ $0\in M_{\ell },$then it must be transversally flat. The latter is best possible. 

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3. Carleson’s $$\varepsilon ^{2}$$ conjecture in higher dimensions

   https://doi.org/10.1007/s00222-025-01337-w  

   Fleschler, Ian; Tolsa, Xavier; Villa, Michele (July 2025, Inventiones mathematicae)

   Abstract In this paper we prove a higher dimensional analogue of Carleson’s$$\varepsilon ^{2}$$ ${\epsilon }^2$conjecture. Given two arbitrary disjoint Borel sets$$\Omega ^{+},\Omega ^{-}\subset \mathbb{R}^{n+1}$$ ${\Omega }^{+},{\Omega }^{-}\subset R^{n+1}$, and$$x\in \mathbb{R}^{n+1}$$ $x\in R^{n+1}$,$$r>0$$ $r>0$, we denote$$ \varepsilon _{n}(x,r) := \frac{1}{r^{n}}\, \inf _{H^{+}} \mathcal{H}^{n} \left ( ((\partial B(x,r)\cap H^{+}) \setminus \Omega ^{+}) \cup (( \partial B(x,r)\cap H^{-}) \setminus \Omega ^{-})\right ), $$ ${\epsilon }_n(x,r):=\frac{1}{r^n}\underset{H^{+}}{inf}{H}^n\left(\right(\left(\partial B\right(x,r)\cap H^{+})\setminus {\Omega }^{+})\cup (\left(\partial B\right(x,r)\cap H^{-})\setminus {\Omega }^{-}\left)\right),$where the infimum is taken over all open affine half-spaces$$H^{+}$$ $H^{+}$such that$$x \in \partial H^{+}$$ $x\in \partial H^{+}$and we define$$H^{-}= \mathbb{R}^{n+1} \setminus \overline{H^{+}}$$ $H^{-}=R^{n+1}\setminus \overline{H^{+}}$. Our first main result asserts that the set of points$$x\in \mathbb{R}^{n+1}$$ $x\in R^{n+1}$where$$ \int _{0}^{1} \varepsilon _{n}(x,r)^{2} \, \frac{dr}{r}< \infty $$ ${\int }_0^1{\epsilon }_n{(x,r)}^2\frac{dr}{r}<\infty$is$$n$$ $n$-rectifiable. For our second main result we assume that$$\Omega ^{+}$$ ${\Omega }^{+}$,$$\Omega ^{-}$$ ${\Omega }^{-}$are open and that$$\Omega ^{+}\cup \Omega ^{-}$$ ${\Omega }^{+}\cup {\Omega }^{-}$satisfies the capacity density condition. For each$$x \in \partial \Omega ^{+} \cup \partial \Omega ^{-}$$ $x\in \partial {\Omega }^{+}\cup \partial {\Omega }^{-}$and$$r>0$$ $r>0$, we denote by$$\alpha ^{\pm }(x,r)$$ ${\alpha }^{\pm }(x,r)$the characteristic constant of the (spherical) open sets$$\Omega ^{\pm }\cap \partial B(x,r)$$ ${\Omega }^{\pm }\cap \partial B(x,r)$. We show that, up to a set of$$\mathcal{H}^{n}$$ $H^n$measure zero,$$x$$ $x$is a tangent point for both$$\partial \Omega ^{+}$$ $\partial {\Omega }^{+}$and$$\partial \Omega ^{-}$$ $\partial {\Omega }^{-}$if and only if$$ \int _{0}^{1} \min (1,\alpha ^{+}(x,r) + \alpha ^{-}(x,r) -2) \frac{dr}{r} < \infty . $$ ${\int }_0^{1}min(1,{\alpha }^{+}(x,r)+{\alpha }^{-}(x,r)-2)\frac{dr}{r}<\infty .$The first result is new even in the plane and the second one improves and extends to higher dimensions the$$\varepsilon ^{2}$$ ${\epsilon }^2$conjecture of Carleson. 

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4. Restricted spaces of holomorphic sections vanishing along subvarieties

   https://doi.org/10.1007/s00209-025-03706-w  

   Coman, Dan; Marinescu, George; Nguyên, Viêt-Anh (May 2025, Mathematische Zeitschrift)

   Abstract LetXbe a compact normal complex space of dimensionnandLbe a holomorphic line bundle onX. Suppose that$$\Sigma =(\Sigma _1,\ldots ,\Sigma _\ell )$$ $\Sigma =({\Sigma }_1,\dots ,{\Sigma }_{\ell })$is an$$\ell $$ $\ell$-tuple of distinct irreducible proper analytic subsets ofX,$$\tau =(\tau _1,\ldots ,\tau _\ell )$$ $\tau =({\tau }_1,\dots ,{\tau }_{\ell })$is an$$\ell $$ $\ell$-tuple of positive real numbers, and let$$H^0_0(X,L^p)$$ $H_0^0(X,L^p)$be the space of holomorphic sections of$$L^p:=L^{\otimes p}$$ $L^p:=L^{\otimes p}$that vanish to order at least$$\tau _jp$$ ${\tau }_{j}p$along$$\Sigma _j$$ ${\Sigma }_j$,$$1\le j\le \ell $$ $1\le j\le \ell$. If$$Y\subset X$$ $Y\subset X$is an irreducible analytic subset of dimensionm, we consider the space$$H^0_0 (X|Y, L^p)$$ $H_0^0\left(X\right|Y,L^p)$of holomorphic sections of$$L^p|_Y$$ $L^p{|}_Y$that extend to global holomorphic sections in$$H^0_0(X,L^p)$$ $H_0^0(X,L^p)$. Assuming that the triplet$$(L,\Sigma ,\tau )$$ $(L,\Sigma ,\tau )$is big in the sense that$$\dim H^0_0(X,L^p)\sim p^n$$ $dim{H}_0^0(X,L^p)\sim p^n$, we give a general condition onYto ensure that$$\dim H^0_0(X|Y,L^p)\sim p^m$$ $dim{H}_0^0\left(X\right|Y,L^p)\sim p^m$. WhenLis endowed with a continuous Hermitian metric, we show that the Fubini-Study currents of the spaces$$H^0_0(X|Y,L^p)$$ $H_0^0\left(X\right|Y,L^p)$converge to a certain equilibrium current onY. We apply this to the study of the equidistribution of zeros inYof random holomorphic sections in$$H^0_0(X|Y,L^p)$$ $H_0^0\left(X\right|Y,L^p)$as$$p\rightarrow \infty $$ $p\to \infty$. 

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

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

   Greenfeld, Rachel; Tao, Terence (December 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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