Congruence classes and maximal nonbases

关于Nathanson的一个与最小渐近基和最大渐近非基相关的问题

On a problem of Nathanson related to minimal asymptotic bases and maximal asymptotic nonbases

本文证明对任意$h\ge2$，自然数集可划分为一个$h$阶最小渐近基和一个$h$阶最大渐近非基，解决了Nathanson在1974年提出的开放问题。

与非有理Eisenstein极大理想相关的尖点子群

Cuspidal subgroups associated with non-rational Eisenstein maximal ideals

本文推广了Ramanujan型Eisenstein同余式到一般水平N，证明了在非平方自由水平下非有理Eisenstein极大理想对应的Ribet猜想推广成立。

同余模与Wiles缺陷的行列式环的最大余子式

Congruence modules and Wiles defects of determinantal rings of maximal minors

研究行列式环最大余子式的同余模与Wiles缺陷，通过余子式大小分析其结构特性。

Maximal Unipotent Monodromy, congruences "à la Lucas" and Algebraic independence

Let $f(z)$ be in $1+z\mathbb{Q}[[z]]$ and $\mathcal{S}$ be an infinite set of prime numbers such that, for all $p\in\mathcal{S}$, we can reduce $f(z)$ modulo $p$. We let $f(z)_{\mid p}$ denote the reduction of $f(z)$ modulo $p$. Generally, when $f(z)$ is D-finite, $f(z)_{\mid p}$ is algebraic over $\mathbb{F}_p(z)$. It turns out that if $f(z)$ is a solution of a polynomial of the form $X-A_p(z)X^{p^l}$, we can use this type of equations to obtain results of transcendence and algebraic independence over $\mathbb{Q}(z)$. In the present paper, we look for conditions on the differential operators …

A maximal extension of the Bloom-Maynard bound for sets with no square differences

We show that if $h\in\mathbb{Z}[x]$ is a polynomial of degree $k$ such that the congruence $h(x)\equiv0\pmod{q}$ has a solution for every positive integer $q$, then any subset of $\{1,2,\ldots,N\}$ with no two distinct elements with difference of the form $h(n)$, with $n$ positive integer, has density at most $(\log N)^{-c\log\log\log N}$, for some constant $c$ that depends only on $k$. This improves on the best bound in the literature, due to Rice, and generalizes a recent result of Bloom and Maynard.

Imaginärquadratische Einbettung von Ordnungen rationaler Quaternionenalgebren, und die nichtzyklischen endlichen Untergruppen der Bianchi-Gruppen

Let k be an imaginary quadratic number field, let F be a rational quaternion algebra and M an extension of F as a quaternion k-algebra. We are going to classify the F-orders which arise as an intersection of F with a maximal M-order; and we are going to prove that the discriminant of such an intersection determines uniquely the isomorphism type of the corresponding maximal M-order. Building on this, we are going to relate this intersection to the intersection of a second rational quaternion algebra F' in M with a second maximal M-order. This allows us to determine whether the Bianchi group ove…