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1. 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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2. Violent Nonlinear Collapse in the Interior of Charged Hairy Black Holes

   https://doi.org/10.1007/s00205-024-02038-z  

   Van_de_Moortel, Maxime (October 2024, Archive for Rational Mechanics and Analysis)

   Abstract We construct a new one-parameter family, indexed by$$\epsilon $$ $ϵ$ , of two-ended, spatially-homogeneous black hole interiors solving the Einstein–Maxwell–Klein–Gordon equations with a (possibly zero) cosmological constant$$\Lambda $$ $\Lambda$ and bifurcating off a Reissner–Nordström-(dS/AdS) interior ($$\epsilon =0$$ $ϵ=0$ ). For all small$$\epsilon \ne 0$$ $ϵ\ne 0$ , we prove that, although the black hole is charged, its terminal boundary is an everywhere-spacelikeKasner singularity foliated by spheres of zero radiusr. Moreover, smaller perturbations (i.e. smaller$$|\epsilon |$$ $\left|ϵ\right|$ ) aremore singular than larger ones, in the sense that the Hawking mass and the curvature blow up following a power law of the form$$r^{-O(\epsilon ^{-2})}$$ $r^{-O\left({ϵ}^{-2}\right)}$ at the singularity$$\{r=0\}$$ $\{r=0\}$ . This unusual property originates from a dynamical phenomenon—violent nonlinear collapse—caused by the almost formation of a Cauchy horizon to the past of the spacelike singularity$$\{r=0\}$$ $\{r=0\}$ . This phenomenon was previously described numerically in the physics literature and referred to as “the collapse of the Einstein–Rosen bridge”. While we cover all values of$$\Lambda \in \mathbb {R}$$ $\Lambda \in R$ , the case$$\Lambda <0$$ $\Lambda <0$ is of particular significance to the AdS/CFT correspondence. Our result can also be viewed in general as a first step towards the understanding of the interior of hairy black holes. 

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3. Ramanujan’s partition generating functions modulo $$\ell $$

   https://doi.org/10.1007/s11139-025-01241-0  

   Bringmann, Kathrin; Craig, William; Ono, Ken (November 2025, The Ramanujan Journal)

   Abstract For the partition functionp(n), Ramanujan proved the striking identities$$\begin{aligned} \begin{aligned} \mathcal {P}_5(q):=\sum _{n\ge 0} p(5n+4)q^n&=5\prod _{n\ge 1} \frac{\left( q^5;q^5\right) _{\infty }^5}{(q;q)_{\infty }^6},\\ \mathcal {P}_{7}(q):=\sum _{n\ge 0} p(7n+5)q^n&=7\prod _{n\ge 1}\frac{\left( q^7;q^7\right) _{\infty }^3}{(q;q)_{\infty }^4}+49q \prod _{n\ge 1}\frac{\left( q^7;q^7\right) _{\infty }^7}{(q;q)_{\infty }^8}, \end{aligned} \end{aligned}$$ $\begin{array}{c}\begin{array}{cc}{P}_5\left(q\right):=\sum _{n\ge 0}p(5n+4)q^n& =5\prod _{n\ge 1}\frac{{\left(q^5,;,q^5\right)}_{\infty }^5}{{(q;q)}_{\infty }^6},\\ P_7\left(q\right):=\sum _{n\ge 0}p(7n+5)q^n& =7\prod _{n\ge 1}\frac{{\left(q^7,;,q^7\right)}_{\infty }^3}{{(q;q)}_{\infty }^4}+49q\prod _{n\ge 1}\frac{{\left(q^7,;,q^7\right)}_{\infty }^7}{{(q;q)}_{\infty }^8},\end{array}\end{array}$ where$$(q;q)_{\infty }:=\prod _{n\ge 1}(1-q^n).$$ ${(q;q)}_{\infty }:={\prod }_{n\ge 1}(1-q^n).$ As these identities imply his celebrated congruences modulo 5 and 7, it is natural to seek, for primes$$\ell \ge 5,$$ $\ell \ge 5,$ closed form expressions of the power series$$ \mathcal {P}_{\ell }(q):=\sum _{n\ge 0} p(\ell n-\delta _{\ell })q^n\pmod {\ell }, $$ $P_{\ell }\left(q\right):=\sum _{n\ge 0}p(\ell n-{\delta }_{\ell })q^n(mod\ell ),$ where$$\delta _{\ell }:=\frac{\ell ^2-1}{24}.$$ ${\delta }_{\ell }:=\frac{{\ell }^2-1}{24}.$ In this paper, we prove that$$ \mathcal {P}_{\ell }(q)\equiv c_{\ell } \dfrac{\mathcal {T}_{\ell }(q)}{\left( q^\ell ; q^\ell \right) _\infty } \pmod {\ell }, $$ $P_{\ell }\left(q\right)\equiv c_{\ell }\frac{T_{\ell }\left(q\right)}{{\left(q^{\ell },;,q^{\ell }\right)}_{\infty }}(mod\ell ),$ where$$c_{\ell }\in \mathbb Z$$ $c_{\ell }\in Z$ is explicit and$${\mathcal {T}}_{\ell }(q)$$ $T_{\ell }\left(q\right)$ is the generating function for the Hecke traces of$$\ell $$ $\ell$ -ramified values of special Dirichlet series for weight$$\ell -1$$ $\ell -1$ cusp forms on$$\textrm{SL}_2(\mathbb Z)$$ ${\text{SL}}_2\left(Z\right)$ . This is a new proof of Ramanujan’s congruences modulo 5, 7, and 11, as there are no nontrivial cusp forms of weight 4, 6, and 10. 

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4. On a Traveling Salesman Problem for Points in the Unit Cube

   https://doi.org/10.1007/s00453-024-01257-w  

   Balogh, József; Clemen, Felix Christian; Dumitrescu, Adrian (September 2024, Algorithmica)

   Abstract LetXbe ann-element point set in thek-dimensional unit cube$$[0,1]^k$$ ${[0,1]}^k$ where$$k \ge 2$$ $k\ge 2$ . According to an old result of Bollobás and Meir (Oper Res Lett 11:19–21, 1992) , there exists a cycle (tour)$$x_1, x_2, \ldots , x_n$$ $x_1,x_2,\dots ,x_n$ through thenpoints, such that$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} \le c_k$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}\le c_k$ , where$$|x-y|$$ $|x-y|$ is the Euclidean distance betweenxandy, and$$c_k$$ $c_k$ is an absolute constant that depends only onk, where$$x_{n+1} \equiv x_1$$ $x_{n+1}\equiv x_1$ . From the other direction, for every$$k \ge 2$$ $k\ge 2$ and$$n \ge 2$$ $n\ge 2$ , there existnpoints in$$[0,1]^k$$ ${[0,1]}^k$ , such that their shortest tour satisfies$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} = 2^{1/k} \cdot \sqrt{k}$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}=2^{1/k}·\sqrt{k}$ . For the plane, the best constant is$$c_2=2$$ $c_2=2$ and this is the only exact value known. Bollobás and Meir showed that one can take$$c_k = 9 \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=9{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$ for every$$k \ge 3$$ $k\ge 3$ and conjectured that the best constant is$$c_k = 2^{1/k} \cdot \sqrt{k}$$ $c_k=2^{1/k}·\sqrt{k}$ , for every$$k \ge 2$$ $k\ge 2$ . Here we significantly improve the upper bound and show that one can take$$c_k = 3 \sqrt{5} \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=3\sqrt{5}{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$ or$$c_k = 2.91 \sqrt{k} \ (1+o_k(1))$$ $c_k=2.91\sqrt{k}(1+o_k\left(1\right))$ . Our bounds are constructive. We also show that$$c_3 \ge 2^{7/6}$$ $c_3\ge 2^{7/6}$ , which disproves the conjecture for$$k=3$$ $k=3$ . Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás–Meir conjecture is proposed. 

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5. 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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