Multiplicative Properties of Hilbert Cubes

Monochromatic Hilbert cubes and arithmetic progressions

The Van der Waerden number $W(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$--colored there exists a monochromatic arithmetic progression of length $k$. Similarly, the Hilbert cube number $h(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$--colored there exists a monochromatic affine $k$--cube, that is, a set of the form$$\left\{x_0 + \sum_{b \in B} b : B \subseteq A\right\}$$ for some $|A|=k$ and $x_0 \in \mathbb{Z}$. We show the following relation between the Hilbert cube number and the Van der Waerden number. Let $k \geq 3$ be an integer. Then for every $ε>0$, …

Upperbound for Dimension of Hilbert Cubes contained in the Quadratic Residues of $\mathbb{F_p}$

We consider the problem of bounding the dimension of Hilbert cubes in a finite field $\mathbb{F_p}$ that does not contain any primitive roots. We show that the dimension of such Hilbert cubes is $O_ε(p^{1/8+ε})$ for any $ε> 0$, matching what can be deduced from the classical Burgess estimate in the special case when the Hilbert cube is an arithmetic progression. We also consider the dual problem of bounding the dimension of multiplicative Hilbert cubes avoiding an interval.

基于平方条件的筛法及其在算术集中的Hilbert立方体应用

Sieving with square conditions and applications to Hilbert cubes in arithmetic sets

本文探讨了Gallagher筛法在模素数平方下的应用局限，并提出通过转换余数信息以提升筛法效果，同时研究了Hilbert立方体在不同乘法定义集中的最大维度，得出了相关上界。

整数平方集中存在无限多个三维希尔伯特立方体

There are infinitely many Hilbert cubes of dimension 3 in the set of squares

本文证明了在整数平方集中存在至少O(N^{1/8})个三维希尔伯特立方体，并证明了相关比例集在正实数中稠密。

Random matrix-valued multiplicative functions and linear recurrences in Hilbert-Schmidt norms of random matrices

We introduce the notion of a random matrix-valued multiplicative function, generalizing Rademacher random multiplicative functions to matrices. We provide an asymptotic for the second moment based on a linear recurrence property for Hilbert-Schmidt norms of sucessive products of random matrices. Moreover, we provide upper bounds for the higher even moments related to the generalized joint spectral radius.

The multiplicative Hilbert matrix

It is observed that the infinite matrix with entries $(\sqrt{mn}\log (mn))^{-1}$ for $m, n\ge 2$ appears as the matrix of the integral operator $\mathbf{H}f(s):=\int_{1/2}^{+\infty}f(w)(ζ(w+s)-1)dw$ with respect to the basis $(n^{-s})_{n\ge 2}$; here $ζ(s)$ is the Riemann zeta function and $H$ is defined on the Hilbert space ${\mathcal H}^2_0$ of Dirichlet series vanishing at $+\infty$ and with square-summable coefficients. This infinite matrix defines a multiplicative Hankel operator according to Helson's terminology or, alternatively, it can be viewed as a bona fide (small) Hankel operator o…

Multiplicative functions arising from the study of mutually unbiased bases

We embed the somewhat unusual multiplicative function, which was serendipitously discovered in 2010 during a study of mutually unbiased bases in the Hilbert space of quantum physics, into two families of multiplicative functions that we construct as generalizations of that particular example. In addition, we report yet another multiplicative function, which is also suggested by that example; it can be used to express the squarefree part of an integer in terms of an exponential sum.

A uniform Weyl bound for L-functions of Hilbert modular forms

We establish a Weyl-type subconvexity of $L(\tfrac{1}{2},f)$ for spherical Hilbert newforms $f$ with level ideal $\mathfrak{N}^2$, in which $\mathfrak{N}$ is required to be cube-free, and at any prime ideal $\mathfrak{p}$ with $\mathfrak{p}^2 \mid \mathfrak{N}$ the local representation generated by $f$ is not supercuspidal. The proof exploits a distributional version of Motohashi's formula over number fields developed by the first author, as well as Katz's work on hypergeometric sums over finite fields in the language of $\ell$-adic cohomology.

多元整数Chebyshev问题

The multivariate integer Chebyshev problem

多元整数Chebyshev问题是在$\C^d.$中的紧集上寻找最小化上确界范数整系数多项式。我们在一般集上研究这个问题，但特别注意乘积集，如立方体和多圆盘。我们还建立了一个多元模拟的Hilbert-Fekete上界的整数切比雪夫常数，这取决于空间的维数。对于复平面上的单变量多项式，我们的估计与Hilbert-Fekete的结果一致.