  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}}|n\alpha -p|\leq \psi(n) \end{equation*} has infinitely many
coprime solutions $(n,p)\in\N\times \Z$ 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}|n\alpha-p|\leq \psi(n) \end{equation*} has infinitely many
coprime solutions $(n,p)\in\N\times \Z$.
