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).

Speaker's profile: Professor Sanju Velani is mainly engaged in research in the fields of number theory, fractal geometry, dynamical systems, and traversal theory, and has achieved multiple breakthrough results. He proposed the contraction target problem, established the principle of mass transfer, and promoted the vigorous development of metric number theory and fractal geometry. He has published 6 papers in the four major mathematical journals (including 3 in Annals of Mathematics, 2 in Inventions Mathematicae, and 1 in Acta Mathematica) and has received a £ 1.6 million grant from the UK EPSRC project. The research team led by him at the University of York in the UK is the most renowned team in the world in the field of Diophantine approximation research.