This means the following: take a point (α, β) in the plane, and then consider the sequence of points For each of these, multiply the distance to the closest line with integer x-coordinate by the distance to the closest line with integer y-coordinate.
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. in the little-o notation.
It is known that this would follow from a result in the geometry of numbers, about the minimum on a non-zero lattice point of a product of three linear forms in three real variables: the implication was shown in 1955 by Cassels and Swinnerton-Dyer.
This in turn is a special case of a general conjecture of Margulis on Lie groups.
Borel showed in 1909 that the exceptional set of real pairs (α,β) violating the statement of the conjecture is of Lebesgue measure zero.
These results imply that non-trivial pairs satisfying the conjecture exist: indeed, given a real number α such that