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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. Increasing subsequences, matrix loci and Viennot shadows

   https://doi.org/10.1017/fms.2024.75  

   Rhoades, Brendon (January 2024, Forum of Mathematics, Sigma)

   Abstract Let$${\mathbf {x}}_{n \times n}$$be an$$n \times n$$matrix of variables, and let$${\mathbb {F}}[{\mathbf {x}}_{n \times n}]$$be the polynomial ring in these variables over a field$${\mathbb {F}}$$. We study the ideal$$I_n \subseteq {\mathbb {F}}[{\mathbf {x}}_{n \times n}]$$generated by all row and column variable sums and all products of two variables drawn from the same row or column. We show that the quotient$${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$$admits a standard monomial basis determined by Viennot’s shadow line avatar of the Schensted correspondence. As a corollary, the Hilbert series of$${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$$is the generating function of permutations in$${\mathfrak {S}}_n$$by the length of their longest increasing subsequence. Along the way, we describe a ‘shadow junta’ basis of the vector space ofk-local permutation statistics. We also calculate the structure of$${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$$as a graded$${\mathfrak {S}}_n \times {\mathfrak {S}}_n$$-module. 

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3. Intersective sets for sparse sets of integers

   https://doi.org/10.1017/etds.2024.73  

   BIENVENU, PIERRE-YVES; GRIESMER, JOHN T; LE, ANH N; LÊ, THÁI HOÀNG (May 2025, Ergodic Theory and Dynamical Systems)

   Abstract For$$E \subset \mathbb {N}$$, a subset$$R \subset \mathbb {N}$$isE-intersectiveif for every$$A \subset E$$having positive relative density,$$R \cap (A - A) \neq \varnothing $$. We say thatRischromatically E-intersectiveif for every finite partition$$E=\bigcup _{i=1}^k E_i$$, there existsisuch that$$R\cap (E_i-E_i)\neq \varnothing $$. When$$E=\mathbb {N}$$, we recover the usual notions of intersectivity and chromatic intersectivity. We investigate to what extent the known intersectivity results hold in the relative setting when$$E = \mathbb {P}$$, the set of primes, or other sparse subsets of$$\mathbb {N}$$. Among other things, we prove the following: (1) the set of shifted Chen primes$$\mathbb {P}_{\mathrm {Chen}} + 1$$is both intersective and$$\mathbb {P}$$-intersective; (2) there exists an intersective set that is not$$\mathbb {P}$$-intersective; (3) every$$\mathbb {P}$$-intersective set is intersective; (4) there exists a chromatically$$\mathbb {P}$$-intersective set which is not intersective (and therefore not$$\mathbb {P}$$-intersective). 

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4. Equality Conditions for the Fractional Superadditive Volume Inequalities

   https://doi.org/10.1007/s00454-024-00672-8  

   Meyer, Mark (July 2025, Discrete & Computational Geometry)

   Abstract While studying set function properties of Lebesgue measure, F. Barthe and M. Madiman proved that Lebesgue measure is fractionally superadditive on compact sets in$$\mathbb {R}^n$$ $R^n$. In doing this they proved a fractional generalization of the Brunn–Minkowski–Lyusternik (BML) inequality in dimension$$n=1$$ $n=1$. In this paper we will prove the equality conditions for the fractional superadditive volume inequalites for any dimension. The non-trivial equality conditions are as follows. In the one-dimensional case we will show that for a fractional partition$$(\mathcal {G},\beta )$$ $(G,\beta )$and nonempty sets$$A_1,\dots ,A_m\subseteq \mathbb {R}$$ $A_1,\cdots ,A_m\subseteq R$, equality holds iff for each$$S\in \mathcal {G}$$ $S\in G$, the set$$\sum _{i\in S}A_i$$ ${\sum }_{i\in S}{A}_i$is an interval. In the case of dimension$$n\ge 2$$ $n\ge 2$we will show that equality can hold if and only if the set$$\sum _{i=1}^{m}A_i$$ ${\sum }_{i=1}^m{A}_i$has measure 0. 

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5. The geometric distribution of Selmer groups of elliptic curves over function fields

   https://doi.org/10.1007/s00208-022-02429-1  

   Feng, Tony; Landesman, Aaron; Rains, Eric M. (September 2022, Mathematische Annalen)

   Abstract Fix a positive integernand a finite field$${\mathbb {F}}_q$$ $F_q$. We study the joint distribution of the rank$${{\,\mathrm{rk}\,}}(E)$$ $\mathrm{rk}\left(E\right)$, then-Selmer group$$\text {Sel}_n(E)$$ ${\text{Sel}}_n\left(E\right)$, and then-torsion in the Tate–Shafarevich group Equation missing<#comment/>asEvaries over elliptic curves of fixed height$$d \ge 2$$ $d\ge 2$over$${\mathbb {F}}_q(t)$$ $F_q\left(t\right)$. We compute this joint distribution in the largeqlimit. We also show that the “largeq, then large height” limit of this distribution agrees with the one predicted by Bhargava–Kane–Lenstra–Poonen–Rains. 

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