Iskander Aliev
We give a sharpened form of Siegel Lemma's w. r. t. the maximum norm. This implies a new lower bound on the greatest element of a sum-distinct set of positive integers (Erdös-Moser problem). The main tools are Minkowski's theorem on successive minima and the Busemann theorem from convex geometry.
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Padé approximations and Siegel's lemma are widely used tools in Diophantine approximation theory. This work has evolved from the attempts to improve Baker-type linear independence measures, either by using the Bombieri-Vaaler version of Siegel's lemma to sharpen the estimates of Padé-type approximations, or by finding completely explicit expressions for the yet unknown 'twin type' Hermite-Padé approximations. The appropriate homogeneous matrix equation representing both methods has an $M \times (L+1)$ coefficient matrix, where $M \le L$. The homogeneous solution vectors of this matrix equation…
On a new absolute version of Siegel's lemma
本文给出数域上Siegel引理的一个绝对版本，仅用线性代数构造满足稀疏性条件的小高度基向量，并给出所有子空间高度的非平凡上界。
Let $K$ be a number field, and let $W$ be a subspace of $K^N$, $N \geq 1$. Let $V_1,...,V_M$ be subspaces of $K^N$ of dimension less than dimension of $W$. We prove the existence of a point of small height in $W \setminus \bigcup_{i=1}^M V_i$, providing an explicit upper bound on the height of such a point in terms of heights of $W$ and $V_1,...,V_M$. Our main tool is a counting estimate we prove for the number of points of a subspace of $K^N$ inside of an adelic cube. As corollaries to our main result we derive an explicit bound on the height of a non-vanishing point for a decomposable form a…
Linear independence of odd zeta values using Siegel's lemma
本文利用西格尔引理证明奇数ζ值的线性独立性，并改进了Rivoal在2001年的下界结果。
Consider a real matrix $Θ$ consisting of rows $(θ_{i,1},\ldots,θ_{i,n})$, for $1\leq i\leq m$. The problem of making the system linear forms $x_{1}θ_{i,1}+\cdots+x_{n}θ_{i,n}-y_{i}$ for integers $x_{j},y_{i}$ small naturally induces an ordinary and a uniform exponent of approximation, denoted by $w(Θ)$ and $\widehat{w}(Θ)$ respectively. For $m=1$, a sharp lower bound for the ratio $w(Θ)/\widehat{w}(Θ)$ was recently established by Marnat and Moshchevitin. We give a short, new proof of this result upon a hypothesis on the best approximation integer vectors associated to $Θ$. Our conditional resu…
In this appendix we present an expanded version of Section 4 of our paper arXiv:1404.4596, including the proofs of all of the technical lemmas.
Let $p$ be a prime, and let $Γ=\Sp_g(\Z)$ be the Siegel modular group of genus $g$. We study $p$-adic families of zeta functions and Siegel modular forms. $L$-functions of Siegel modular forms are described in terms of motivic $L$-functions attached to $\Sp_g$, and their analytic properties are given. Critical values for the spinor $L$-functions and $p$-adic constructions are discussed. Rankin's lemma of higher genus is established. A general conjecture on a lifting from $ GSp_{2m} \times GSp_{2m}$ to $GSp_{4m}$ (of genus $g=4m$) is formulated. Constructions of $p$-adic families of Siegel modu…
In their celebrated paper "On Siegel's Lemma", Bombieri and Vaaler found an upper bound on the height of integer solutions of systems of linear Diophantine equations. Calculating the bound directly, however, requires exponential time. In this paper, we present the bound in a different form that can be computed in polynomial time. We also give an elementary (and arguably simpler) proof for the bound.
Given two Siegel eigenforms of different weights, we determine explicit sets of Hecke eigenvalues for the two forms that must be distinct. In degree two, and under some additional conditions, we determine explicit sets of Fourier coefficients of the two forms that must be distinct.
In this paper we follow the approach of Bertrand-Beukers (and of later work of Bertrand), based on differential Galois theory, to prove a very general version of Shidlovsky's lemma that applies to Pad{é} approximation problems at several points, both at functional and numerical levels (i.e., before and after evaluating at a specific point). This allows us to obtain a new proof of the Ball-Rivoal theorem on irrationality of infinitely many values of Riemann zeta function at odd integers, inspired by the proof of the Siegel-Shidlovsky theorem on values of E-functions: Shidlovsky's lemma is used …
Let $F$ and $G$ be Siegel cusp forms for $\Sp_4(\Z)$ and weights $k_1, k_2$ respectively. Also let $F$ and $G$ be Hecke eigenforms lying in distinct eigen spaces. Further suppose that neither $F$ nor $G$ is a Saito-Kurokawa lift. In this article, we study simultaneous arithmetic behaviour of Hecke eigenvalues of these Hecke eigenforms.
Consider a rational point on an elliptic curve under an isogeny. Suppose that the action of Galois partitions the set of its pre-images into n orbits. It is shown that all such points above a certain height have their denominator divisible by at least n distinct primes. This generalizes Siegel's theorem and more recent results of Everest et al. For multiplication by a prime l, it is shown that if n>1 then either the point is l times a rational point or the elliptic curve emits a rational l-isogeny.
Let $X\cong \operatorname{SL}_2(\mathbb R)/\operatorname{SL}_2(\mathbb Z)$ be the space of unimodular lattices in $\mathbb R^2$, and for any $r\ge 0$ denote by $K_r\subset X$ the set of lattices such that all its nonzero vectors have supremum norm at least $e^{-r}$. These are compact nested subset{s} of $X$, with $K_0 = {\bigcap}_{r}K_r$ being the union of two closed horocycles. We use an explicit second moment formula for the Siegel transform of the indicator functions of squares in $\mathbb R^2$ centered at the origin to derive an asymptotic formula for the volume of sets $K_r$ as $r\to 0$. …
Hilbert surfaces, modular forms, and Siegel-Veech constants
本文研究了翻译表面中普林姆特征形式定位的希伯特-维克常数，证明其在不同定位间一致，并揭示了相关希伯特模曲面的欧拉特征关系。
The Chowla conjecture and Landau-Siegel zeroes
本文在Landau-Siegel零点存在的条件下，对Chowla猜想中的相关和式建立了非平凡界，改进了先前结果。
Intrinsic Diophantine approximation by rationals of height with a bounded number of distinct prime factors
本文研究了由具有有限不同素因子数的有理数逼近的自相似集的Hausdorff维数，修正了前一版本中引理3.8的证明错误。
Suppose that some harmonic analysis arguments have been invoked to show that the indicator function of a set of residue classes modulo some integer has a large Fourier coefficient. To get information about the structure of the set of residue classes, we then need a certain type of complementary result. A solution to this problem was given by Gregory Freiman in 1961, when he proved a lemma which relates the value of an exponential sum with the distribution of summands in semi-circles of the unit circle in the complex plane. Since then, Freiman's Lemma has been extended by several authors. Rathe…
We establish new measures of linear independence of logarithms on commutative algebraic groups in the so-called \emph{rational case}. More precisely, let k be a number field and v_{0} be an arbitrary place of k. Let G be a commutative algebraic group defined over k and H be a connected algebraic subgroup of G. Denote by Lie(H) its Lie algebra at the origin. Let u\in Lie(G(C_{v_{0}})) a logarithm of a point p\in G(k). Assuming (essentially) that p is not a torsion point modulo proper connected algebraic subgroups of G, we obtain lower bounds for the distance from u to Lie(H)\otimes_{k} C_{v_{0}…
The main result in Y.~O.~Hamidoune's paper "Adding Distinct Congruence Classes" ({\em Combin.~Probab.~Comput.}~{\bf 7} (1998) 81-87) is as follows: If $S$ is a generating subset of a cyclic group $G$ such that $0 \not \in S$ and $|S| \geq 5$, then the number of sums of the subsets of $S$ is at least $\min (|G|, 2|S| )$. Unfortunately, argument of the author, who, sadly, passed away in 2011, relies on a lemma whose proof is incorrect; in fact, the lemma is false for all cyclic groups of even order. In this short note we point out this mistake, correct the proof, and discuss why the main result …
Rademacher expansion of a Siegel modular form for ${\cal N}= 4$ counting
本文利用 Igusa 尖点 Siegel 模形式 $Φ_{10}$ 的辛对称性，构造了细粒度的 Rademacher 型展开，将杂化弦理论中 $1/4$ BPS 态简并度表示为极点留数的正则化求和，并揭示了其与 $1/η^{24}$ 傅里叶系数的连分数结构关系。
Quantitative bounds for sets lacking polynomial progressions with shifted prime difference
该研究证明了移位参数限制在移位素数集合的多项式进展的定量Szemerédi型定理，改进了线性情形结果，不同次数非线性构型的定量界为首创，结合了整数多项式构型与素数Gowers一致性界等方法。
Every quasiperfect number has at least eight distinct prime factors
本文通过三个中学代数引理消除了准完全数下界的阻碍，结合机器计算首次完整证明准完全数若存在则至少有8个不同素因子，填补了此前研究的覆盖缺口。
The Five Distance Theorem For An Arbitrary Norm
该论文证明了二维严格凸范数下，幺模格上克罗内克序列的最近邻距离数至多为五，解决了对应ℓₚ范数的猜想界，扩展了相关格论论证。
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