DSNT Seminar: Winning of inhomogeneous bad for curves

Abstract: Badly approximable vectors are one of the central topics in Diophantine approximation due to their connection with homogeneous dynamics and other problems in number theory. Moreover, the intersection of the sets of several weighted badly approximable vectors is the content of Schmidt's conjecture, now a theorem by Badziahin, Pollington, and Velani (Annals 2011).  The holy grail of the theory of badly approximable vectors is to show the `winning property' of these sets inside curves and fractals. In a recent breakthrough, Beresnevich, Nesharim, and Yang  (Duke 2022) showed the `absolute winning property' of weighted bad inside analytic nonplanar curves. We extend their work from homogeneous to the inhomogeneous set-up. As a consequence, our result also establishes the inhomogeneous Schmidt's conjecture in arbitrary dimensions. This is a joint work with Liyang Shao. I will not assume any prior knowledge in this topic. 

This is a seminar in our seminar series on Dynamical Systems and Number Theory (DSNT).