Partial Results
Progress has been made in showing that the exceptional set of real pairs (α,β) violating the statement of the conjecture must be small. Manfred Einsiedler, Anatole Katok and Elon Lindenstrauss have shown that it must have Hausdorff dimension zero; and in fact is a union of countably many compact sets of box-counting dimension zero. The result was proved by using measure classification theorem for diagonalizable actions of higher-rank groups, and an isolation theorem proved by Lindenstrauss and Barak Weiss.
Read more about this topic: Littlewood Conjecture
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