Diophantine approximation is the study of approximating real numbers by rational numbers. For example, one can ask whether a real number x is ψ-approximable; that is, whether there are infinitely many rationals p/q satisfying |x-p/q|<ψ(q)/q for a given monotonic function ψ. A century ago, Khintchine discovered a remarkable dichotomy for the Lebesgue measure of the set of ψ-approximable numbers. Since then, Khintchine’s theorem has been extended in various directions, including inhomogeneous approximation and higher-dimensional generalizations. In this talk, I will introduce the Allen-Ramírez conjecture on removing the monotonicity condition from the inhomogeneous Khintchine-Groshev theorem and discuss a recent proof of the conjecture in the case (n,m)=(2,1).

강연시간 : 16:40-17:10