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1. Uniform Lech’s inequality

   https://doi.org/10.1090/proc/16304  

   Ma, Linquan; Smirnov, Ilya (February 2023, Proceedings of the American Mathematical Society)

   Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d\ge <\#comment/>2$ . We prove that if $e\left({\stackrel{^<\#comment/>}{R}}_{red}\right)>1$ , then the classical Lech’s inequality can be improved uniformly for all $\mathfrak{m}$ -primary ideals, that is, there exists $\mathrm{\epsilon <\#comment/>}>0$ such that $e\left(I\right)\le <\#comment/>d!\left(e\right(R)-<\#comment/>\mathrm{\epsilon <\#comment/>})\mathrm{\ell <\#comment/>}\left(R/I\right)$ for all $\mathfrak{m}$ -primary ideals $I\subseteq <\#comment/>R$ . This answers a question raised by Huneke, Ma, Quy, and Smirnov [Adv. Math. 372 (2020), pp. 107296, 33]. We also obtain partial results towards improvements of Lech’s inequality when we fix the number of generators of $I$ . 

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2. Products of normal subsets

   https://doi.org/10.1090/tran/8960  

   Larsen, Michael; Shalev, Aner; Tiep, Pham (February 2024, Transactions of the American Mathematical Society)

   In this paper we consider which families of finite simple groups $G$ have the property that for each $\mathrm{ϵ<\#comment/>}>0$ there exists $N>0$ such that, if $|G|\ge <\#comment/>N$ and $S,T$ are normal subsets of $G$ with at least $\mathrm{ϵ<\#comment/>}|G|$ elements each, then every non-trivial element of $G$ is the product of an element of $S$ and an element of $T$ . We show that this holds in a strong and effective sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form ${\mathrm{P}\mathrm{S}\mathrm{L}}_n\left(q\right)$ where $q$ is fixed and $n\to <\#comment/>\mathrm{\infty <\#comment/>}$ . However, in the case $S=T$ and $G$ alternating this holds with an explicit bound on $N$ in terms of $\mathrm{ϵ<\#comment/>}$ . Related problems and applications are also discussed. In particular we show that, if $w_1,w_2$ are non-trivial words, $G$ is a finite simple group of Lie type of bounded rank, and for $g\in <\#comment/>G$ , $P_{w_1\left(G\right),w_2\left(G\right)}\left(g\right)$ denotes the probability that $g_1{g}_2=g$ where $g_i\in <\#comment/>w_i\left(G\right)$ are chosen uniformly and independently, then, as $|G|\to <\#comment/>\mathrm{\infty <\#comment/>}$ , the distribution $P_{w_1\left(G\right),w_2\left(G\right)}$ tends to the uniform distribution on $G$ with respect to the $L^{\mathrm{\infty <\#comment/>}}$ norm. 

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3. Berge pancyclic hypergraphs

   https://doi.org/10.1090/proc/17370  

   Bailey, Teegan; Li, Yupei; Luo, Ruth (May 2026, Proceedings of the American Mathematical Society)

   An $n$ -vertex graph $G$ ispancyclicif for every $\mathrm{\ell <\#comment/>}$ , $3\le <\#comment/>\mathrm{\ell <\#comment/>}\le <\#comment/>n$ , $G$ contains a cycle of length $\mathrm{\ell <\#comment/>}$ . We consider an analog for hypergraphs. An $r$ -uniform hypergraph is a set of vertices and a set of edges in which every edge contains $r$ vertices. A Berge cycle of length $\mathrm{\ell <\#comment/>}$ in a hypergraph is an alternating sequence of $\mathrm{\ell <\#comment/>}$ distinct vertices and $\mathrm{\ell <\#comment/>}$ distinct edges $v_1,e_1,v_2,\dots <\#comment/>,v_{\mathrm{\ell <\#comment/>}},e_{\mathrm{\ell <\#comment/>}},v_1$ such that $\{v_i,v_{i+1}\}\subseteq <\#comment/>e_i$ for all $i$ , with indices taken modulo $\mathrm{\ell <\#comment/>}$ . We similarly call a hypergraphBerge pancyclicif it contains Berge cycles of every possible length. For sufficiently large $n$ we prove a sharp minimum degree condition guaranteeing Berge pancyclicity in $r$ -uniform hypergraphs where $3\le <\#comment/>r\le <\#comment/>$ $\lfloor <\#comment/>(n-<\#comment/>1)/2\rfloor <\#comment/>-<\#comment/>2$ . This bound is optimal for all such combinations of $n$ and $r$ . 

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4. On the Boundary Behavior of Mass-Minimizing Integral Currents

   https://doi.org/10.1090/memo/1446  

   De_Lellis, Camillo; De_Philippis, Guido; Hirsch, Jonas; Massaccesi, Annalisa (November 2023, Memoirs of the American Mathematical Society)

   Let $\mathrm{\Sigma <\#comment/>}$ be a smooth Riemannian manifold, $\mathrm{\Gamma <\#comment/>}\subset <\#comment/>\mathrm{\Sigma <\#comment/>}$ a smooth closed oriented submanifold of codimension higher than $2$ and $T$ an integral area-minimizing current in $\mathrm{\Sigma <\#comment/>}$ which bounds $\mathrm{\Gamma <\#comment/>}$ . We prove that the set of regular points of $T$ at the boundary is dense in $\mathrm{\Gamma <\#comment/>}$ . Prior to our theorem the existence of any regular point was not known, except for some special choice of $\mathrm{\Sigma <\#comment/>}$ and $\mathrm{\Gamma <\#comment/>}$ . As a corollary of our theorem we answer to a question in Almgren’sAlmgren’s big regularity paperfrom 2000 showing that, if $\mathrm{\Gamma <\#comment/>}$ is connected, then $T$ has at least one point $p$ of multiplicity $\frac{1}{2}$ , namely there is a neighborhood of the point $p$ where $T$ is a classical submanifold with boundary $\mathrm{\Gamma <\#comment/>}$ ; we generalize Almgren’s connectivity theorem showing that the support of $T$ is always connected if $\mathrm{\Gamma <\#comment/>}$ is connected; we conclude a structural result on $T$ when $\mathrm{\Gamma <\#comment/>}$ consists of more than one connected component, generalizing a previous theorem proved by Hardt and Simon in 1979 when $\mathrm{\Sigma <\#comment/>}={\mathbb{R}}^{m+1}$ and $T$ is $m$ -dimensional. 

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5. Super-polynomial accuracy of one dimensional randomized nets using the median of means

   https://doi.org/10.1090/mcom/3791  

   Pan, Zexin; Owen, Art (March 2023, Mathematics of Computation)

   Let $f$ be analytic on $[0,1]$ with $|f^{\left(k\right)}\left(1/2\right)|⩽<\#comment/>A{\mathrm{\alpha <\#comment/>}}^{k}k!$ for some constants $A$ and $\mathrm{\alpha <\#comment/>}>2$ and all $k⩾<\#comment/>1$ . We show that the median estimate of $\mathrm{\mu <\#comment/>}={\int <\#comment/>}_0^{1}f\left(x\right)\mathrm{d}x$ under random linear scrambling with $n=2^m$ points converges at the rate $O\left(n^{-<\#comment/>c\log <\#comment/>\left(n\right)}\right)$ for any $c>3\log <\#comment/>\left(2\right)/{\mathrm{\pi <\#comment/>}}^2\approx <\#comment/>0.21$ . We also get a super-polynomial convergence rate for the sample median of $2k-<\#comment/>1$ random linearly scrambled estimates, when $k/m$ is bounded away from zero. When $f$ has a $p$ ’th derivative that satisfies a $\mathrm{\lambda <\#comment/>}$ -Hölder condition then the median of means has error $O\left(n^{-<\#comment/>(p+\mathrm{\lambda <\#comment/>})+\mathrm{ϵ<\#comment/>}}\right)$ for any $\mathrm{ϵ<\#comment/>}>0$ , if $k\to <\#comment/>\mathrm{\infty <\#comment/>}$ as $m\to <\#comment/>\mathrm{\infty <\#comment/>}$ . The proof techniques use methods from analytic combinatorics that have not previously been applied to quasi-Monte Carlo methods, most notably an asymptotic expression from Hardy and Ramanujan on the number of partitions of a natural number. 

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