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
Famous quotes containing the words partial and/or results:
“The one-eyed man will be King in the country of the blind only if he arrives there in full possession of his partial faculties—that is, providing he is perfectly aware of the precise nature of sight and does not confuse it with second sight ... nor with madness.”
—Angela Carter (1940–1992)
“Life and language are alike sacred. Homicide and verbicide—that is, violent treatment of a word with fatal results to its legitimate meaning, which is its life—are alike forbidden.”
—Oliver Wendell Holmes, Sr. (1809–1894)