Some refined results on mixed Littlewood conjecture for pseudo-absolute values
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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 the sharp criterion such that for almost every $alpha$ the inequality \begin{equation*} |n|_{mathcal{D}}|nalpha -p|leq psi(n) end{equation*} has infinitely many coprime solutions $(n,p)inN imes $ 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}|nalpha-p|leq psi(n) end{equation*} has infinitely many coprime solutions $(n,p)inN imes $.
