The Littlewood conjecture states that $\liminf_{n\to\infty}||n\alpha|| ||n\beta|=0|$ holds for all real numbers $\alpha$ and $\alpha$, where $||\cdot||$ denotes the distance to the nearest integer. There are several other formulations of Littlewood conjecture, including the $p$-adic and mixed Littlewood conjecture. In this talk, I start with an introduction to the history of different versions of Littlewood conjecture.  Then I will present several refined results of mixed Littlewood conjecture for pseudo-absolute values.
Let $\mathcal{D}_1$,$\mathcal{D}_2,\cdots, \mathcal{D}_k$ be $k$ pseudo absolute sequences and define the  $\mathcal{D}$-adic norm $|\cdot|_{\mathcal{D}}:\N\to \{n_k^{-1}:k\ge 0\}$ by $|{n}|_\mathcal{D} = \min\{ n_k^{-1} : n\in  n_k\Z \}.$
Under some minor condition of $\mathcal{D}_1$,$\mathcal{D}_2,\cdots, \mathcal{D}_k$,  I set up the criteria of 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||\leq \psi(n)
\end{equation*}
has infinitely many solutions for $n\in\N$. Under some minor condition of the pseudo absolute sequence $\mathcal{D}$, I also show that
for any $\alpha\in\R$, $\liminf_{n\to \infty}n|n|_a|n|_\mathcal{D}\|n\alpha\|=0.$