# Meeting Details

Title: A question of Wolfgang Schmidt concerning two dimensional Diophantine approximation Algebra and Number Theory Seminar Andrew D. Pollington, National Science Foundation I will start with a short introduction to Diophantine approximation, how well can real numbers be approximated by rational numbers, and then go on to some recent results concerning higher dimensional questions. In the 1930's Littlewood conjectured that for every pair of real numbers (x,y) inf{q||qx|| ||qy||:q in N}=0. Here ||x|| denotes the distance of x from the nearest integer. This conjecture is still open, although we now know, thanks to work of Einsiedler, Katok and Lindenstrauss, that the set of possible counter examples is a set of Hausdorff dimension 0. In any counter example to the conjecture there must be a constant $c>0$ so that for any pair of non negative real numbers (i,j), with i+j=1, q max (||qx||^{1/i}, ||qy||^{1/j}) > c for all natural numbers q. We say that such a pair (x,y) belongs to Bad(i,j). It relatively easy to show that for each possible pair (i,j) that Bad(i,j) is non empty, and in fact has full Hausdorff dimension in the plane. The question that Schmidt asked in 1970 was to show that there are pairs which lie in both Bad(1/3,2/3) and in Bad(2/3,1/3). In this talk I will show that the intersection of any finite collection of Bad(i,j) sets is non-empty. This is joint work with Dzmitry Badziahin, now at Durham University, and Sanju Velani at the University of York.