Victor Cuauhtemoc Garcia, Florin Nicolae
Let $f(z)=\sum_{n=1}^{\infty}a(n) e^{2πi nz}$ be a normalized Hecke eigenform in $S_{2k}^{\text{new}}(Γ_0(N))$ with integer Fourier coefficients. We prove that there exists a constant $C(f)>0$ such that any integer is a sum of at most $C(f)$ coefficients $a(n) $. It holds $C(f)\ll_{\varepsilon,k}N^{\frac{6k-3}{16}+\varepsilon}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We study the rational Bianchi newforms (weight 2, trivial character, with rational Hecke eigenvalues) in the LMFDB that are not associated to elliptic curves, but instead to abelian surfaces with quaternionic multiplication. Two of these examples exhibit a rather special kind of behaviour: we show they arise from twisted base change of a classical newform with nebentypus character of order 4 and eight inner twists.
The sup-norm problem for newforms of large level on $\operatorname{PGL}(n)$
本文研究了大水平新形式在PGL(n)上的上范数问题，通过新的降维理论和Atkin-Lehner算子推广，证明了次下界并基于GRH条件扩展结果。
Benford's Law for Coefficients of Newforms
设$f(z)=\sum_{n=1}^\infty λ_f(n)e^{2πi n z}\in S_{k}^{new}(Γ_0(N))$是$Γ_0(N)$上的偶数权$k\geq2$的归一化Hecke本征形，无需复数乘法。令$\mathbb{P}$表示所有素数的集合。证明了序列$\{λ_f(p)\}_{p\in\mathbb{P}}$在任何基$b\geq2$上都不满足Benford定律.然而，给定基$b\geq2$和基$b$中的数字串$S$，集合\[ A_{λ_f}(b,S):=\{\text{$p$ prime : the first digits of $λ_f(p)$ in base $b$ are given by $S$}\} \]具有等于$\log_b(1+S^{-1})$的对数密度。因此，$\{λ_f(p)\}_{p\in\mathbb{P}}$在对数密度方面遵循本福德定律。这两个结果都依赖于现已证明的Sato-Tate猜想。
Let $F/E$ be a Galois extension of totally real number fields, with Galois group $\mathrm{Gal}(F/E)$. Let $\mathfrak{N}$ be an integral ideal which is $\mathrm{Gal}(F/E)$-invariant, and $k \ge 2$ an integer. In this note, we study the action of $\mathrm{Gal}(F/E)$ on the Hecke orbits of Hilbert newforms of level $\mathfrak{N}$ and weight $k$. We also discuss the geometric counterpart to this action, which is closely related to the notion of abelian varieties potentially of $\mathrm{GL}_2$-type. The two actions have some consequences in relation with Langlands Functoriality. We conclude with an…
Langlands Base Change for GL(2)
设F是一个全实Galois数域。在简单的局部假设下，我们证明了奇水平的每一个经典新形式关于扩张F/Q的基变的存在性。
We prove a bound for the Fourier coefficients of a cusp form of integral weight which is not a newform by computing an explicit orthogonal basis for the space of cusp forms of given integral weight and level. In contrast to previous work on special cases of this problem we use elementary methods for the computation of Petersson products and avoid using the Rankin $L$-function.
Let $\ell \geq 5$ be a prime and let $N$ be a square-free integer prime to $\ell$. For each prime $p$ dividing $N$, let $a_p$ be either $1$ or $-1$. We give sufficient criteria for the existence of a newform $f$ of weight 2 for $Γ_0(N)$ such that the mod $\ell$ Galois representation attached to $f$ is reducible and $U_p f = a_p f$ for primes $p$ dividing $N$. The main techniques used are level raising methods based on an exact sequence due to Ribet.
Paul Erd\H os and additive bases
这是一个调查的一些埃尔德\H os的工作基地在加性数论。
The concept of additive basis has been investigated in the literature for several mathematicians which works with number theorem. Recently, the concept of finitely stable additive basis was introduced. In this note we provide a counterexample of the reciprocal of Theorem 2.2 shown in [Ferreira, L.A.: Finite Stable Additive Basis; Bull. Aust. Math. Soc.]. The idea of the construction of such a counterexample can possibly help the process of finding additive bases of some specific order.
Additive bases arising from functions in a Hardy field
经典的加性基问题是华林问题。它已被推广到整数多项式和非整数幂序列。在本文中，我们将考虑一个更广泛的功能，即功能从一个Hardy域，并表明他们是渐近的基础。
We prove that finite sets of real numbers satisfying $|AA| \leq |A|^{1+ε}$ with sufficiently small $ε> 0$ cannot have small additive bases nor can they be written as a set of sums $B+C$ with $|B|, |C| \geq 2$. The result can be seen as a real analog of the conjecture of Sárközy that multiplicative subgroups of finite fields of prime order are additively irreducible.
A set of non-negative integers A is an additive 2-basis with range n, if its sumset A+A contains 0, 1, ..., n but not n+1. Explicit bases are known with arbitrarily large size |A|=k and $n/k^2 \ge 2/7 > 0.2857$. We present a more general construction and improve the lower bound to $85/294 > 0.2891$.
Let $E$ be a rational elliptic curve, and $K$ be an imaginary quadratic field. In this article we give a method to construct Heegner points when $E$ has a prime bigger than $3$ of additive reduction ramifying in the field $K$. The ideas apply to more general contexts, like constructing Darmon points attached to real quadratic fields which is presented in the appendix.
Additive bases and Niven numbers
在胡利黎曼假设下，证明基$g$的Niven数集是加性基，即每个自然数可表示为至多$C_g$个此类数之和。
An extension of the Erdős-Turán additive base conjecture via generalized circles of partition
本研究推广了分圆概念，并以此为工具证明了Erdős-Turán加法基猜想，推进了数论中相关猜想的证明进展。
A simple proof that any additive basis has only finitely many essential subsets
设$A$为加法基。我们称$A$的“本质子集”为$A$的任何有限子集$P$，使得$A \setminus P$不是可加基，并且$P$是最小的（对于包含顺序）以具有此性质。B. Deschamps和作者指出，任何加性基都只有100多个基本子集（参见“添加剂基中的基本子集”，J.数论，123（2007），第170- 192页“）。本文的目的是给出这个定理的一个简单证明。
Robust additive bases without minimal subbases
该研究构造了一个加法基底集合，其中表示数的下界与log m相关，但不存在最小子集使得其和集包含所有大整数。
On the computation of base-change lifts and lifts of Hida families
本文推导了Hilbert模形式的Hecke特征值公式，证明了基变换提升形式的L函数可分解为扭曲L函数的乘积，并展示了Hida族的基变换提升存在性。
Let $b$ be a numeration base. A $b$-additive Ramanujan-Hardy number $N$ is an integer for which there exists at least an integer $M$, called additive multiplier, such that the product of $M$ and the sum of base $b$ digits of $N$, added to the reversal of the product, gives $N$. We show that for any $b$ there exists an infinity of $b$-additive Ramanujan-Hardy numbers and an infinity of additive multipliers. A $b$-multiplicative Ramanujan-Hardy number $N$ is an integer for which there exists at least an integer $M$, called multiplicative multiplier, such that the product of $M$ and the sum of ba…
In [10] the third author of this paper presented two conjectures on the additive decomposability of the sequence of ''smooth'' (or ''friable'') numbers. Elsholtz and Harper [4] proved (by using sieve methods) the second (less demanding) conjecture. The goal of this paper is to extend and sharpen their result in three directions by using a different approach (based on the theory of $S$-unit equations).
Problems in additive number theory, VII: The structure of additive $h$-bases for $n$
本文研究加性数论中有限整数集作为$h$-基的结构问题，提出了一类适合AI求解的新问题。
Generalized additive bases and difference bases for Cartesian product of finite abelian groups
本文研究有限 Abel 群笛卡尔积 $G^n$ 中的广义 $g$-加法基与 $g$-差基，通过显式构造给出其最小规模的渐近尖锐上界。
A meet-in-the-middle algorithm for finding extremal restricted additive 2-bases
一个值域为n的可加2-基若其最大元素为n/2，则是受限的.在给定长度k的限制2-基中，具有最大值域的基是极值限制的。我们描述了一个算法，发现极值限制2-基地的一个给定的长度，我们列出他们的长度为k = 41。
A new upper bound for finite additive bases
设n（2，k）表示最大整数n，对于该最大整数n存在k个非负整数的集合A，使得和集2A包含{0，1，2，.，n-1}。加性数论中的一个经典问题是求n（2，k）的上界。本文证明了limsup_{k\to\infty} n（2，k）/k^2 \leq 0.4789.
Dense Edge-Magic Graphs and Thin Additive Bases
我们研究s（k，n），A+A的最大尺寸，其中A是[n]的k-子集。加性数论中的一些已知函数可以用s（k，n）表示。例如，我们对s（k，n）的估计意味着拟Sidon集的最大大小的新界限，这是Erdos和Freud [J. Number Th.38（1991）196-205]提出的问题。此外，应用所谓的边魔术标号的图。
Problems in additive number theory, I
加性数论中与和集和差集、更一般的二元线性型以及整数和非负整数的加性基的表示函数有关的问题。
We study a generalization of additive bases into a planar setting. A planar additive basis is a set of non-negative integer pairs whose vector sumset covers a given rectangle. Such bases find applications in active sensor arrays used in, for example, radar and medical imaging. The problem of minimizing the basis cardinality has not been addressed before. We propose two algorithms for finding the minimal bases of small rectangles: one in the setting where the basis elements can be anywhere in the rectangle, and another in the restricted setting, where the elements are confined to the lower left…
Supersequences, rearrangements of sequences, and the spectrum of bases in additive number theory
如果每个足够大的整数都可以表示为A的h个元素之和，则非负整数集合A = {a_n}是h阶渐进基。如果a_n ~ alpha n^h对某个实数alpha > 0，则alpha称为h阶可加特征值。h阶加性谱是由所有h阶加性特征值组成的集合N（h）。证明了存在一个正数eta_h <= 1/h！使得N（h）=（0，eta_h）或N（h）=（0，eta_h]。证明使用的结果建设的超序列的序列与规定的渐近增长，也约重排的无穷序列的渐近性。例如，证明了不存在严格递增的整数序列B = {b_n}使得b_n ~ 2 ^n，且B包含子序列{ b_{n_k} }使得b_{n_k} ~ 3 ^k.
Distinct Matroid Base Weights and Additive Theory
设$M$是集合$E$上的拟阵，$w:E\longrightarrow G$是权函数，其中$G$是循环群.假设$w(E)$满足Pollard条件（即$w(E)-w(E)$的每个非零元素生成$G$），我们得到了不同基权数的公式。如果$|G|$是素数，我们的结果与Schrijver和Seymour的结果一致.我们还描述了等式的情况下，在这个公式。在素数情形下，我们的结果推广了Vozen定理.
Additive Bases in Abelian Groups
设$G$是一个指数为$m$的有限非平凡阿贝尔群，并假设$B_1, ..., B_k$是$G$的生成子集。我们证明了，如果$k>2m \ln \log_2 |G|$，则多集并$B_1\cup...\cup B_k$形成$G$的加性基;即，对于每个$g\in G$存在$A_1\subset B_1, ..., A_k\subset B_k$，使得$g=\sum_{i=1}^k\sum_{a\in A_i} a$。这推广了Alon，Linial和Meshulam关于加法基猜想的一个结果.作为证明猜想的另一个步骤，在$B_1, ..., B_k$是向量空间的有限子集的情况下，我们获得不同值的数量的下界估计，通过$\sum_{i=1}^k \sum_{a\in A_i} a$的形式的总和获得，其中$A_i$在每个$i=1, >..., k$的$B_i$的所有子集上变化。最后，我们建立了一个令人惊讶的加性基猜想和单位立方体的顶点覆盖问题之间的关系，通过平移的格，并提出了一个（强形式）的猜想在覆盖方面的重新制定。
Sharp Threshold Asymptotics for the Emergence of Additive Bases
{0，1，.，n}被称为是{1，2，.，n}如果{1，2，.，n}可以写成j=x+y，x，y在A中，x<=y。如果我们在{0，1，.}中选取每个整数，n}独立地以概率p=p_n趋于0，从而得到一个随机集A，那么我们得到一个2-可加基的概率是多少？当目标和集是[（1-alpha）n，（1+alpha）n]（或等价地[alpha n，（2-alpha）n]）时，我们解决这个问题，其中0<alpha<1。在这两种模型下，使用Poisson近似的Stein-Chen方法，结合Janson不等式，梳理出一个非常尖锐的阈值，出现一个2-加性基。然后给出了k-可加基的推广。
Additive bases in groups
本文研究了一般阿贝尔群中从加性基中去掉一个元素的问题。我们引进类似的经典功能$X$，$S$和$E$（定义在整数的情况下），并获得他们的界限。我们估计的功能$S_G$和$E_G$是有效的一般阿贝尔群$G$，而在$X_G$的情况下，我们表明，不同类型的行为可能会发生依赖于组$G$。
Generalized additive bases, Konig's lemma, and the Erdos-Turan conjecture
设A是一组非负整数。对于每个非负整数n和正整数h，设r_{A}（n，h）表示n的表示形式n = a_1 + a_2 +. + a_h，其中a_1，a_2，.，a_h是A的元素，a_1 \leq a_2 \leq. \leq a_h.无限集A称为h阶基，如果对每个非负整数n，有r_{A}（n，h）\geq 1. Erdos和Turan证明了对每个2阶基A，limsup_{n\to\infty} r_A（n，2）= \infty.本文引入了一类新的可加基和一般的可加问题，其特例是Erdos-Turan猜想。利用Konig关于某些图中存在无限路的引理，证明了这个一般问题等价于一个有关的非负整数的有限集问题.
Answers to two questions posed by Farhi concerning additive bases
设A是N的渐近基，X是A的有限子集，使得A \X仍是渐近基. Farhi最近根据A的阶和与集合X有关的各种参数证明了A \X的阶的一批新的上界。他提出了两个问题，可能改善他的界限。在本说明中，我们回答了这两个问题。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。