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SOME REFINED RESULTS ON THE MIXED LITTLEWOOD CONJECTURE FOR PSEUDO-ABSOLUTE VALUES
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 107 / Issue 1 / August 2019
- Published online by Cambridge University Press:
- 22 August 2018, pp. 91-109
- Print publication:
- August 2019
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- Article
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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 has infinitely many coprime solutions
$$\begin{eqnarray}|n|_{{\mathcal{D}}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$
$(n,p)\in \mathbb{N}\times \mathbb{Z}$ 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 has infinitely many coprime solutions
$$\begin{eqnarray}|n|_{{\mathcal{D}}_{1}}|n|_{{\mathcal{D}}_{2}}\cdots |n|_{{\mathcal{D}}_{k}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$
$(n,p)\in \mathbb{N}\times \mathbb{Z}$.
