Intrinsic Diophantine approximation for overlapping iterated function systems

Abstract In this paper we study a family of limsup sets that are defined using iterated function systems. Our main result is an analogue Khintchine’s theorem for these sets. We then apply to the topic intrinsic Diophantine Approximation on self-similar particular, define new height element $${\mathbb {Q}}^d$$ Q d contained in set terms its eventually periodic representations. For with respect function, obtain detailed description their metric properties. The results hold arbitrary dimensions and without any separation conditions underlying system.