SOME REFINED RESULTS ON THE MIXED LITTLEWOOD CONJECTURE FOR PSEUDO-ABSOLUTE VALUES
SOME REFINED RESULTS ON THE MIXED LITTLEWOOD CONJECTURE FOR PSEUDO-ABSOLUTE VALUES
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DOI：
10.1017/s1446788718000198
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发表时间：
2017-09
影响因子：
0.7
通讯作者：
Wencai Liu
中科院分区：
文献类型：
--
作者：
Wencai Liu
本文研究了带伪绝对值的混合Littlewood猜想。对于任意伪绝对值序列${\mathcal{D}}$，我们得到了一个尖锐的判据，使得对于几乎每一个$\unicode[STIX]{x1D6FC}$不等式$$\begin{eqnarray}|n|_{{\mathcal{D}}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$对于某一单参数族$\unicode[STIX]{x1D713}$都有无穷多个互素解$(n,p)\in \mathbb{N}\times \mathbb{Z}$。此外，在伪绝对值序列${\mathcal{D}}_{1},{\mathcal{D}}_{2},\ldots ,{\mathcal{D}}_{k}$上的一个次要条件下，我们得到了一般序列$\unicode[STIX]{x1D713}(n)$上的一个尖锐判据，使得对于几乎每一个$\unicode[STIX]{x1D6FC}$不等式$$\begin{eqnarray}|n|_{{\mathcal{D}}_{1}}|n|_{{\mathcal{D}}_{2}}\cdots |n|_{{\mathcal{D}}_{k}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$都有无限多个协素解$(n,p)\in \mathbb{N}\times \mathbb{Z}$。
In this paper, we study the mixed Littlewood conjecture with pseudo-absolute values. For any pseudo-absolute-value sequence ${\mathcal{D}}$ , we obtain a sharp criterion such that for almost every $\unicode[STIX]{x1D6FC}$ the inequality $$\begin{eqnarray}|n|_{{\mathcal{D}}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$ has infinitely many coprime solutions $(n,p)\in \mathbb{N}\times \mathbb{Z}$ for a certain one-parameter family of $\unicode[STIX]{x1D713}$ . Also, under a minor condition on pseudo-absolute-value sequences ${\mathcal{D}}_{1},{\mathcal{D}}_{2},\ldots ,{\mathcal{D}}_{k}$ , we obtain a sharp criterion on a general sequence $\unicode[STIX]{x1D713}(n)$ such that for almost every $\unicode[STIX]{x1D6FC}$ the inequality $$\begin{eqnarray}|n|_{{\mathcal{D}}_{1}}|n|_{{\mathcal{D}}_{2}}\cdots |n|_{{\mathcal{D}}_{k}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$ has infinitely many coprime solutions $(n,p)\in \mathbb{N}\times \mathbb{Z}$ .