Speaker: Dmitry Badziahin (University of Sydney)
24 February, 2021 10:00 UTC
Abstract: In the talk we discuss the set $S_n(\lambda)$ of simultaneously $\lambda$-well approximable points in $\mathbb{R}^n$. These are the points $x$ such that the inequality \(\| x - p/q\|_\infty < q^{-\lambda - \epsilon}\) has infinitely many solutions in rational points $p/q$. Investigating the intersection of this set with a suitable manifold comprises one of the most challenging problems in Diophantine approximation. It is known that the structure of this set, especially for large $\lambda$, depends on the manifold. For some manifolds it may be empty, while for others it may have relatively large Hausdorff dimension.
We will concentrate on the case of the Veronese curve $V_n$. We discuss, what is known about the Hausdorff dimension of the set $S_n(\lambda) \cap V_n$ and will talk about the recent results of the speaker and Bugeaud which impose new bounds on that dimension.