Let T be a d × d matrix with integral coefficients. Then T determines a self-map of the d-dimensional torus T d = R d/Z d . 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 ∈ T d such that T n (x) ∈ B(n) for infinitely many n ∈ N. The size of W measured in terms of ddimensional Lebesgue measure (restricted to T d ) and Haudsorff dimension are pretty much well understood. In this talk I explore the situation in which the points x ∈ T d are restricted to a nice subset M (such as an analytic sub-manifold) of T d ; 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{k2 nxk, k3 nxk} < ψ(n) for infinitely many n ∈ N.
There will be a social reception following this event at Koerner's Pub. Please contact Caleb Marshall (cmarshall@math...) to RSVP, or for Zoom information.