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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. 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 (March 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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3. Counting imaginary quadratic fields with an ideal class group of 5-rank at least 2

   https://doi.org/10.1007/s11139-025-01184-6  

   Bartz, Kollin; Levin, Aaron; Thamminana, Aman_Dhruva (August 2025, The Ramanujan Journal)

   Abstract We prove that there are$$\gg \frac{X^{\frac{1}{3}}}{(\log X)^2}$$ $\gg \frac{X^{\frac{1}{3}}}{{(logX)}^2}$imaginary quadratic fieldskwith discriminant$$|d_k|\le X$$ $|d_k|\le X$and an ideal class group of 5-rank at least 2. This improves a result of Byeon, who proved the lower bound$$\gg X^{\frac{1}{4}}$$ $\gg X^{\frac{1}{4}}$in the same setting. We use a method of Howe, Leprévost, and Poonen to construct a genus 2 curveCover$$\mathbb {Q}$$ $Q$such thatChas a rational Weierstrass point and the Jacobian ofChas a rational torsion subgroup of 5-rank 2. We deduce the main result from the existence of the curveCand a quantitative result of Kulkarni and the second author. 

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4. Negative moments of the Riemann zeta-function

   https://doi.org/10.1515/crelle-2023-0091  

   Bui, Hung M; Florea, Alexandra (January 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Assuming the Riemann Hypothesis, we study negative moments of the Riemann zeta-function and obtain asymptotic formulas in certain ranges of the shift in $\zeta \left(s\right)${\zeta(s)}. For example, integrating ${\left|\zeta \left(\frac{1}{2}+\alpha +it\right)\right|}^{-2k}${|\zeta(\frac{1}{2}+\alpha+it)|^{-2k}}with respect totfromTto $2T${2T}, we obtain an asymptotic formula when the shift α is roughly bigger than $\frac{1}{\log T}${\frac{1}{\log T}}and $k<\frac{1}{2}${k<\frac{1}{2}}. We also obtain non-trivial upper bounds for much smaller shifts, as long as $\log \frac{1}{\alpha }\ll \log \log T${\log\frac{1}{\alpha}\ll\log\log T}. This provides partial progress towards a conjecture of Gonek on negative moments of the Riemann zeta-function, and settles the conjecture in certain ranges. As an application, we also obtain an upper bound for the average of the generalized Möbius function. 

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5. Enumeration of Rooted Binary Unlabeled Galled Trees

   https://doi.org/10.1007/s11538-024-01270-8  

   Agranat-Tamir, Lily; Mathur, Shaili; Rosenberg, Noah A. (March 2024, Bulletin of Mathematical Biology)

   Abstract Rooted binarygalledtrees generalize rooted binary trees to allow a restricted class of cycles, known asgalls. We build upon the Wedderburn-Etherington enumeration of rooted binary unlabeled trees withnleaves to enumerate rooted binary unlabeled galled trees withnleaves, also enumerating rooted binary unlabeled galled trees withnleaves andggalls,$$0 \leqslant g \leqslant \lfloor \frac{n-1}{2} \rfloor $$ $0⩽g⩽\lfloor \frac{n-1}{2}\rfloor$. The enumerations rely on a recursive decomposition that considers subtrees descended from the nodes of a gall, adopting a restriction on galls that amounts to considering only the rooted binarynormalunlabeled galled trees in our enumeration. We write an implicit expression for the generating function encoding the numbers of trees for alln. We show that the number of rooted binary unlabeled galled trees grows with$$0.0779(4.8230^n)n^{-\frac{3}{2}}$$ $0.0779(4.{8230}^n)n^{-\frac{3}{2}}$, exceeding the growth$$0.3188(2.4833^n)n^{-\frac{3}{2}}$$ $0.3188(2.{4833}^n)n^{-\frac{3}{2}}$of the number of rooted binary unlabeled trees without galls. However, the growth of the number of galled trees with only one gall has the same exponential order 2.4833 as the number with no galls, exceeding it only in the subexponential term,$$0.3910n^{\frac{1}{2}}$$ $0.3910n^{\frac{1}{2}}$compared to$$0.3188n^{-\frac{3}{2}}$$ $0.3188n^{-\frac{3}{2}}$. For a fixed number of leavesn, the number of gallsgthat produces the largest number of rooted binary unlabeled galled trees lies intermediate between the minimum of$$g=0$$ $g=0$and the maximum of$$g=\lfloor \frac{n-1}{2} \rfloor $$ $g=\lfloor \frac{n-1}{2}\rfloor$. We discuss implications in mathematical phylogenetics. 

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