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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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.