Estimating lower limit in the $p$-adic Littlewood conjecture

We verify that ${\mathrm{lim\hspace{0.17em}inf}}_{q\to \mathrm{\infty }}q\cdot \mathrm{\mid }q{\mathrm{\mid }}_p\cdot \mathrm{\mid }\mathrm{\mid }qx\mathrm{\mid }\mathrm{\mid }<\epsilon$ for all real $x$, small primes $p$ and relatively small $\epsilon$. This result supports the famous $p$-adic Littlewood conjecture which states that the above lower limit is equal to 0 for all $x\in \mathbb{R}$. In particular, the result is established for $p=2$ with $\epsilon =1\mathrm{/}25$. For $3\le p\le 29$, the upper bounds for $\epsilon$ vary, but they are always at most $1\mathrm{/}10$.