Abstract
AbstractIn this paper we study a family of limsup sets that are defined using iterated function systems. Our main result is an analogue of Khintchine’s theorem for these sets. We then apply this result to the topic of intrinsic Diophantine Approximation on self-similar sets. In particular, we define a new height function for an element of$${\mathbb {Q}}^d$$Qdcontained in a self-similar set in terms of its eventually periodic representations. For limsup sets defined with respect to this height function, we obtain a detailed description of their metric properties. The results of this paper hold in arbitrary dimensions and without any separation conditions on the underlying iterated function system.
Funder
Engineering and Physical Sciences Research Council
Publisher
Springer Science and Business Media LLC
Reference40 articles.
1. Allen, D., Bárány, B.: Hausdorff measures of shrinking targets on self-conformal sets. Mathematika 67(4), 807–839 (2021)
2. Baker, S.: An analogue of Khintchine’s theorem for self-conformal sets. Math. Proc. Cambridge Philos. Soc. 167(3), 567–597 (2019)
3. Baker, S.: Approximation properties of $$\beta $$-expansions. Acta Arith. 168, 269–287 (2015)
4. Baker, S.: Approximation properties of $$\beta $$-expansion II. Ergodic Theory Dynam. Syst. 38(5), 1627–1641 (2018)
5. Baker, S.: Overlapping iterated function systems from the perspective of Metric Number Theory. Mem. Amer. Math. Soc. (to appear)