论数域中的最小模问题

全球函数域中Erdős覆盖系统的界

Bounds for Erdős covering systems in global function fields

本文研究了全球函数域中覆盖系统的界，证明了在特定条件下不存在具有乘法次数s的覆盖系统，并改进了之前与李、王和伊合作的研究结果。

关于全球函数域中的埃尔德斯覆盖系统的研究

On Erdős covering systems in global function fields

本文基于 distortion 方法，证明了在特定条件下，全球函数域中不存在具有特定乘数的覆盖系统，尤其指出当 q≥759 时，F_q[x] 上不存在具有不同模数的覆盖系统。

On covering systems of integers

A covering system of the integers is a finite collection of modular residue classes $\{a_m \bmod{m}\}_{m \in S}$ whose union is all integers. Given a finite set $S$ of moduli, it is often difficult to tell whether there is a choice of residues modulo elements of $S$ covering the integers. Hough has shown that if the smallest modulus in $S$ is at least $10^{16}$, then there is none. However, the question of whether there is a covering of the integers with all odd moduli remains open. We consider multiplicative restrictions on the set of moduli to generalize Hough's negative solution to the mini…

Periodicity of power Fibonacci sequences modulus a Fibonacci number

Let ${\mathcal F}=(F_i:i\ge 0)$ be the sequence of Fibonacci numbers, and $j$ and $e$ be non negative integers. We study the periodicity of the power Fibonacci sequences ${\mathcal F}^e(F_j)=(F_i^e\pmod{F_j}: i\ge 0)$. It is shown that for every $j,e\ge 1$ the sequence ${\mathcal F}^e(F_j)$ is periodic and its periodicity is computed. The result was previously known for ${\mathcal F}(F_j)$; that is, for $e=1$. For $e\in \{1, 2\}$, the values of the normalized residues $ρ_i\equiv F_i^e\pmod{F_j}$ with $0\le ρ_i<F_j-1$ are obtained.

Quartic and Octic Characters Modulo n

The average number of primitive quadratic Dirichlet characters of modulus n tends to a constant as n->infty. The same is true for primitive cubic characters. It is therefore surprising that, as n->infty, the average number of primitive quartic characters of modulus n grows with ln(n), and that the average number of primitive octic characters of modulus n grows with ln(n)^2. Leading coefficients in the asymptotic expressions are also computed.

关于具有不同模的覆盖系统的第$j$小模

On the $j$-th smallest modulus of a covering system with distinct moduli

本研究在Balister等人对Hough关于不同模覆盖系统最小模有界性的证明基础上，推导出具有不同模的极小覆盖系统的第$j$小模的上界为$\exp(cj^2/\log(j+1))$，其中$c$为正绝对常数。

数域的序的射线类群与射线类域

Ray class groups and ray class fields for orders of number fields

本文定义了数域中任意序与任意射线类模相关的射线类群，证明其射线类域存在性，揭示其与经典射线类群/环类群的联系，给出序和模变换的正合序列，并刻画射线类域的子域性质与素数分裂行为。

Voronoi summation for ${\rm GL}_n$: collusion between level and modulus

We investigate the Voronoi summation problem for ${\rm GL}_n$ in the level aspect for $n\geq 2$. Of particular interest are those primes at which the level and modulus are jointly ramified - a common occurrence in analytic number theory when using techniques such as the Petersson trace formula. Building on previous legacies, our formula stands as the most general of its kind; in particular we extend the results of Ichino-Templier. We also give (classical) refinements of our formula and study the $p$-adic generalisations of the Hankel transform.

Fault-Tolerant Modular Reconstruction of Rational Numbers

In this paper we present two efficient methods for reconstructing a rational number from several residue-modulus pairs, some of which may be incorrect. One method is a natural generalization of that presented by Wang, Guy and Davenport in \cite{WGD1982} (for reconstructing a rational number from \textit{correct} modular images), and also of an algorithm presented in \cite{Abb1991} for reconstructing an \textit{integer} value from several residue-modulus pairs, some of which may be incorrect.

Characterization of the numbers which satisfy the height reducing property

Let $α$ be a complex number. We show that there is a finite subset $F$ of the ring of the rational integers $\mathbb{Z}$, such that $F\left[ α\right] =\mathbb{Z}\left[ α\right]$, if and only if $α$ is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus greater than one. This completes the answer to a question, on the numbers satisfying the height reducing property, posed in [3].

Erdős covering systems

A covering system is a finite collection of arithmetic progressions whose union is the set of integers. The study of these objects was initiated by Erdős in 1950, and over the following decades he asked many questions about them. Most famously, he asked whether there exist covering systems with distinct moduli whose minimum modulus is arbitrarily large. This problem was resolved in 2015 by Hough, who showed that in any such system the minimum modulus is at most $10^{16}$. The purpose of this note is to give a gentle exposition of a simpler and stronger variant of Hough's method, which was rece…

Rational exponential sums over the divisor function

We consider a problem posed by Shparlinski, of giving nontrivial bounds for rational exponential sums over the arithmetic function $τ(n)$, counting the number of divisors of $n$. This is done using some ideas of Sathe concerning the distribution in residue classes of the function $ω(n)$, counting the number of prime factors of $n$, to bring the problem into a form where, for general modulus, we may apply a bound of Bourgain concerning exponential sums over subgroups of finite abelian groups and for prime modulus some results of Korobov and Shkredov.

The structure and number of Erdős covering systems

Introduced by Erdős in 1950, a covering system of the integers is a finite collection of arithmetic progressions whose union is the set $\mathbb{Z}$. Many beautiful questions and conjectures about covering systems have been posed over the past several decades, but until recently little was known about their properties. Most famously, the so-called minimum modulus problem of Erdős was resolved in 2015 by Hough, who proved that in every covering system with distinct moduli, the minimum modulus is at most $10^{16}$. In this paper we answer another question of Erdős, asked in 1952, on the number o…

Nori motives of curves with modulus and Laumon 1-motives

Let $k$ be a number field. We describe the category of Laumon 1-isomotives over $k$ as the universal category in the sense of Nori associated with a quiver representation built out of smooth proper $k$-curves with two disjoint effective divisors and a notion of $H^1_\dR$ for such "curves with modulus". This result extends and relies on the theorem of J. Ayoub and L. Barbieri-Viale that describes Deligne's category of 1-isomotives in terms of Nori's Abelian category of motives.

A Multi-layer Recursive Residue Number System

We present a method to increase the dynamical range of a Residue Number System (RNS) by adding virtual RNS layers on top of the original RNS, where the required modular arithmetic for a modulus on any non-bottom layer is implemented by means of an RNS Montgomery multiplication algorithm that uses the RNS on the layer below. As a result, the actual arithmetic is deferred to the bottom layer. The multiplication algorithm that we use is based on an algorithm by Bajard and Imbert, extended to work with pseudo-residues (remainders with a larger range than the modulus). The resulting Recursive Resid…

On singular moduli that are S-units

Recently Yu. Bilu, P. Habegger and L. Kühne proved that no singular modulus can be a unit in the ring of algebraic integers. In this paper we study for which sets S of prime numbers there is no singular modulus that is an S-units. Here we prove that when the set S contains only primes congruent to 1 modulo 3 then no singular modulus can be an S-unit. We then give some remarks on the general case and we study the norm factorizations of a special family of singular moduli.

简单快速的错误定位多项式求根算法：模搜索

Simple and fast Algorithm for Finding Roots of Error-Locator Polynomials: Modulus Search

提出一种通过嵌套循环搜索有限域多项式根的模搜索算法，利用改进的Chien搜索和模除法，适用于大型多项式，比Chien搜索快10-100倍。