Formulation and Explanation
This means the following: take a point (α,β) in the plane, and then consider the sequence of points
- (2α,2β), (3α,3β), ... .
For each of these consider the closest lattice point, as determined by multiplying the distance to the closest line with integer x-coordinate by the distance to the closest line with integer y-coordinate. This product will certainly be at most 1/4. The conjecture makes no statement about whether this sequence of values will converge; it typically does not, in fact. The conjecture states something about the limit inferior, and says that there is a subsequence for which the distances decay faster than the reciprocal, i.e.
- o(1/n)
in the little-o notation.
Read more about this topic: Littlewood Conjecture
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