Let T be a d × d matrix with integral coefficients. Then T determines a self-map of the d-dimensional torus Td = Rd/Zd. Choose for each natural number n a ball B(n) in X and suppose that B(n + 1) has smaller radius than B(n) for all n. Thus the ball shrinks as n increases. Now let W be the set of points x ∈ Td such that Tn(x) ∈ B(n) for infinitely many n ∈ N. The size of W measured in terms of ddimensional Lebesgue measure (restricted to Td) and Hausdorff dimension are pretty much well understood. In this talk I explore the situation in which the points x ∈ Td are restricted to a nice subset M (such as an analytic sub-manifold) of Td, that is, the points of interest are functionally dependent. I will essentially concentrate on the situation when d = 2, T has first row (2, 0) and second row (0, 3) and M is the diagonal. In this special case, given a decreasing function ψ, understanding the shrinking target set W ∩M is equivalent to understanding the set of x ∈ [0, 1] such that max{∥2nx∥, ∥3nx∥} < ψ(n) for infinitely many n ∈ N.

This is joint work with Bing Li (South China University of Technology), Lingmin Liao (UPEC) and Evgeniy Zorin (York).

报告人简介：Sanju Velani教授主要从事数论、分形几何、动力系统与遍历理论等领域的研究，取得了多项突破性成果。他提出了收缩靶问题、建立了质量转移原理，推动了度量数论与分形几何的蓬勃发展。他在数学四大杂志中发表论文6篇（其中Annals of Mathematics 3篇、Inventiones Mathematicae 2篇、Acta Mathematica 1篇），曾获得160万英镑的英国EPSRC项目资助。他领导的英国约克大学的研究团队，是世界上丢番图逼近研究方向最富盛名的团队。