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Estimating lower limit in the $p$-adic Littlewood conjecture
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abstract
We verify that $\liminf_{q\to\infty} q\cdot |q|_p\cdot ||qx||<\epsilon$ 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 $\epsilon=1/25$. For $3\le p\le 29$, the upper bounds for $\epsilon$ vary, but they are always at most $1/10$.
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Cited by 1 Pith paper
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Some Bounds Related to the $2$-adic Littlewood Conjecture
For the limsup version of the 2-adic Littlewood conjecture, the paper proves the uniform bound must be at least 5; for the max version, a computer-assisted search raises the bound to 15.
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