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SOME REFINED RESULTS ON THE MIXED LITTLEWOOD CONJECTURE FOR PSEUDO-ABSOLUTE VALUES

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  22 August 2018

WENCAI LIU
WENCAI LIU*
Affiliation:
Department of Mathematics, University of California Irvine, California 92697-3875, USA email [email protected]
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Abstract

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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 $(n,p)\in \mathbb{N}\times \mathbb{Z}$.

Keywords

Littlewood conjecturepseudo-absolute valuesdiophantine approximation

MSC classification

Primary: 11K60: Diophantine approximation
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 107 , Issue 1 , August 2019 , pp. 91 - 109
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The author was supported by the AMS-Simons Travel Grant 2016–2018 and NSF DMS-1700314. This research was also partially supported by NSF DMS-1401204.

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