For almost every real number x, the inequality |x-p/q|<1/q^a has finitely many solutions if and only if a>2. By Roth's theorem, any irrational algebraic number x also satisfies this property, so that from that point of view, algebraic numbers and random numbers behave similarly. We shall present some generalizations of this phenomenon, related to old and new results in diophantine approximation in dependent quantitities, such as Sprindzuk's solution to Baker's conjecture, and Kleinbock and Margulis's solution to Sprindzuk's conjecture. Proofs will use ideas of Kleinbock and Margulis on analysis on the space of lattices in R^d, as well as Schmidt's subspace theorem. Time permitting, we shall also give an application to diophantine approximation on spheres.
Thu, 04/05/2017 - 10:00 to 11:00