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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 (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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2. Calderón–Zygmund theory for strongly coupled linear system of nonlocal equations with Hölder-regular coefficient

   https://doi.org/10.1515/acv-2024-0005  

   Mengesha, Tadele; Schikorra, Armin; Seesanea, Adisak; Yeepo, Sasikarn (November 2024, Advances in Calculus of Variations)

   Abstract We extend the Calderón–Zygmund theory for nonlocal equations tostrongly coupled system of linear nonlocal equations ${\mathcal{ℒ}}_A^{s}u=f${\mathcal{L}^{s}_{A}u=f}, where the operator ${\mathcal{ℒ}}_A^s${\mathcal{L}^{s}_{A}}is formally given by ${\mathcal{ℒ}}_A^{s}u={\int }_{{ℝ}^n}\frac{A(x,y)}{{\left|x-y\right|}^{n+2s}}\frac{\left(x-y\right)\otimes \left(x-y\right)}{{\left|x-y\right|}^2}\left(u\left(x\right)-u\left(y\right)\right)𝑑y.$\mathcal{L}^{s}_{A}u=\int_{\mathbb{R}^{n}}\frac{A(x,y)}{|x-y|^{n+2s}}\frac{(x-%y)\otimes(x-y)}{|x-y|^{2}}(u(x)-u(y))\,dy. For $0<s<1${0<1}and $A:{ℝ}^n\times {ℝ}^n\to ℝ${A:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}}taken to be symmetric and serving asa variable coefficient for the operator, the system under consideration is the fractional version of the classical Navier–Lamé linearized elasticity system. The study of the coupled system of nonlocal equations is motivated by its appearance in nonlocal mechanics, primarily in peridynamics. Our regularity result states that if $A(\cdot ,y)${A(\,\cdot\,,y)}is uniformly Holder continuous and ${inf}_{x\in {ℝ}^n}A(x,x)>0${\inf_{x\in\mathbb{R}^{n}}A(x,x)>0}, then for $f\in L_{\mathrm{loc}}^p${f\in L^{p}_{\rm loc}}, for $p\ge 2${p\geq 2}, the solution vector $u\in H_{\mathrm{loc}}^{2s-\delta ,p}${u\in H^{2s-\delta,p}_{\rm loc}}for some $\delta \in (0,s)${\delta\in(0,s)}. 

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3. Evaluation codes arising from symmetric polynomials

   https://doi.org/10.1007/s10623-025-01637-5  

   Gatti, Barbara; Korchmáros, Gábor; Nagy, Gábor P; Pallozzi_Lavorante, Vincenzo; Schulte, Gioia (May 2025, Designs, Codes and Cryptography)

   Abstract Datta and Johnsen (Des Codes Cryptogr 91:747–761, 2023) introduced a new family of evaluation codes in an affine space of dimension$$\ge 2$$ $\ge 2$over a finite field$${\mathbb {F}}_q$$ $F_q$where linear combinations of elementary symmetric polynomials are evaluated on the set of all points with pairwise distinct coordinates. In this paper, we propose a generalization by taking low dimensional linear systems of symmetric polynomials. Computation for small values of$$q=7,9$$ $q=7,9$shows that carefully chosen generalized Datta–Johnsen codes$$\left[ \frac{1}{2}q(q-1),3,d\right] $$ $\left(\frac{1}{2},q,(q-1),,,3,,,d\right)$have minimum distancedequal to the optimal value minus 1. 

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4. 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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5. An Allard-type boundary regularity theorem for $2d$ minimizing currents at smooth curves with arbitrary multiplicity

   https://doi.org/10.1007/s10240-024-00144-y  

   De Lellis, Camillo; Nardulli, Stefano; Steinbrüchel, Simone (February 2024, Publications mathématiques de l'IHÉS)

   Abstract We consider integral area-minimizing 2-dimensional currents$$T$$ $T$in$$U\subset \mathbf {R}^{2+n}$$ $U\subset R^{2+n}$with$$\partial T = Q\left [\!\![{\Gamma }\right ]\!\!]$$ $\partial T=Q〚\Gamma 〛$, where$$Q\in \mathbf {N} \setminus \{0\}$$ $Q\in N\setminus \left\{0\right\}$and$$\Gamma $$ $\Gamma$is sufficiently smooth. We prove that, if$$q\in \Gamma $$ $q\in \Gamma$is a point where the density of$$T$$ $T$is strictly below$$\frac{Q+1}{2}$$ $\frac{Q+1}{2}$, then the current is regular at$$q$$ $q$. The regularity is understood in the following sense: there is a neighborhood of$$q$$ $q$in which$$T$$ $T$consists of a finite number of regular minimal submanifolds meeting transversally at$$\Gamma $$ $\Gamma$(and counted with the appropriate integer multiplicity). In view of well-known examples, our result is optimal, and it is the first nontrivial generalization of a classical theorem of Allard for$$Q=1$$ $Q=1$. As a corollary, if$$\Omega \subset \mathbf {R}^{2+n}$$ $\Omega \subset R^{2+n}$is a bounded uniformly convex set and$$\Gamma \subset \partial \Omega $$ $\Gamma \subset \partial \Omega$a smooth 1-dimensional closed submanifold, then any area-minimizing current$$T$$ $T$with$$\partial T = Q \left [\!\![{\Gamma }\right ]\!\!]$$ $\partial T=Q〚\Gamma 〛$is regular in a neighborhood of $$\Gamma $$ $\Gamma$. 

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