Intrinsic approximation for fractals defined by rational iterated function systems: Mahler's research suggestion

In this paper, we consider intrinsic Diophantine approximation in the sense of Mahler ['Some suggestions for further research', Bull. Aust. Math. Soc. 29 (1984) 101-108.] on the Cantor set and similar fractals. We begin by obtaining a Dirichlet-type theorem for the limit set of a rational iterated function system. Next, we investigate the rigidity of this result by applying a random affine transformation to such a fractal and determining the intrinsic Diophantine theory of the image fractal. The final two sections concern the optimality of the Dirichlet-type theorem established at the beginning. The first of these seeks to show optimality in the sense that any proof using the same method as ours cannot prove a better approximation exponent, in a precise sense. This is done by introducing a new height function on the rationals intrinsic to the fractal and studying the Diophantine properties of points on the fractal with respect to this new height function. In the final section, we use a result of S. Ramanujan to give a lower bound on the periods of rationals which could cause exceptions to the optimality of the approximation exponent (this time with the usual height function). We give a heuristic argument suggesting that there are only finitely many rationals with periods so large if this is true, then the approximation exponent is optimal for the Cantor set.