The arithmetic of thin orbits

We consider the local-to-global question for orbits of thin groups/semigroups. We will discuss Apollonian circle packings, continued fractions, and some related problems. In the Apollonian case, we ask about the integers which occur as curvatures in a packing. We observe that they satisfy certain congruence restrictions, and ask whether all sufficiently large integers otherwise occur. In the case of continued fractions, we consider variants of Zaremba's conjecture on the rationals with bounded continued fractions. Joint work includes work with Haag, Kertzer, and Rickards.