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

Abstract

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 $\alpha$ the inequality \begin{equation*} |n|_{\mathcal{D}}|nα-p|\leq ψ(n) \end{equation*} has infinitely many coprime solutions $(n,p)\in \mathbb{N}\times \mathbb{Z}$ for a certain one-parameter family of $\psi$. 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 $\psi (n)$ such that for almost every $\alpha$ the inequality \begin{equation*} |n|_{\mathcal{D}_1}|n|_{\mathcal{D}_2}\cdots |n|_{\mathcal{D}_k}|nα-p|\leq ψ(n) \end{equation*} has infinitely many coprime solutions $(n,p)\in \mathbb{N}\times \mathbb{Z}$.

J. Aust. Math. Soc. to appear

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• Two Erdos problems on lacunary sequences: Chromatic number and Diophantine approximation
• On multiplicatively badly approximable numbers
• The Duffin-Schaeffer type conjectures in various local fields