The mixed Littlewood conjecture for pseudo-absolute values

Some refined results on mixed Littlewood conjecture for pseudo-absolute values

In this paper, we study the mixed Littlewood conjecture with pseudo-absolute values. For any pseudo absolute value sequence $\mathcal{D}$, we obtain the sharp criterion such that for almost every $α$ the inequality \begin{equation*} |n|_{\mathcal{D}}|nα-p|\leq ψ(n) \end{equation*} has infinitely many coprime solutions $(n,p)\in\N\times \Z$ for a certain one-parameter family of $ψ$. Also under minor condition on pseudo absolute value sequences $\mathcal{D}_1$,$\mathcal{D}_2,\cdots, \mathcal{D}_k$, we obtain a sharp criterion on general sequence $ψ(n)$ such that for almost every $α$ the inequali…

一致Littlewood猜想反例的获胜性质

Winning property of counterexamples to Uniform Littlewood's Conjecture

本文证明了BFK25提出的一致Littlewood猜想反例集合是超平面绝对获胜的，且在二维实数空间中具有满Hausdorff维数。

On the complexity of a putative counterexample to the $p$-adic Littlewood conjecture

Let $|| \cdot ||$ denote the distance to the nearest integer and, for a prime number $p$, let $| \cdot |_p$ denote the $p$-adic absolute value. In 2004, de Mathan and Teulié asked whether $\inf_{q \ge 1} \, q \cdot || q α|| \cdot | q |_p = 0$ holds for every badly approximable real number $α$ and every prime number $p$. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number $α$ grows too rapidly or too slowly, then their conjecture is true for the pair $(α, p)$ with $p$ an arbitrary prime.

On the $t$-adic Littlewood Conjecture

The $p$-adic Littlewood Conjecture due to De Mathan and Teulié asserts that for any prime number $p$ and any real number $α$, the equation $$\inf_{|m|\ge 1} |m|\cdot |m|_p\cdot |\langle mα\rangle|\, =\, 0 $$ holds. Here, $|m|$ is the usual absolute value of the integer $m$, $|m|_p$ its $p$-adic absolute value and $ |\langle x\rangle|$ denotes the distance from a real number $x$ to the set of integers. This still open conjecture stands as a variant of the well-known Littlewood Conjecture. In the same way as the latter, it admits a natural counterpart over the field of formal Laurent series $\ma…

Littlewood的$L^α$平坦定理的一个推广，其中$α>0$

A generalization of Littlewood's $L^α$ flat theorem, $α>0$

本研究推广了Littlewood的$L^α$平坦性准则，证实了$L^α$-Littlewood等多个猜想，证明Barker序列有限，还构造出Mahler平坦的Gauss-Fresnel多项式，为相关定理提供新证法，成果可应用于数论与动力系统谱理论。

伪绝对值：基础

Pseudo-absolute values: foundations

本文引入伪绝对值概念，推广通常绝对值，赋予其紧豪斯多夫拓扑，并定义伪值域上的解析空间，作为扎里斯基-黎曼空间的解析对应物。