[bibtex key=”BlairHamkinsOBryant2020:Representing-ordinal-numbers-with-arithmetically-interesting-sets-of-real-numbers”]

Abstract. For a real number 𝑥 and set of natural numbers 𝐴, define 𝑥 ∗𝐴 ={𝑥⁢𝑎mod ⁡1 ∣𝑎 ∈𝐴} ⊆[0,1). We consider relationships between 𝑥, 𝐴, and the order-type of 𝑥 ∗𝐴. For example, for every irrational 𝑥 and order-type 𝛼, there is an 𝐴 with 𝑥 ∗𝐴 ≃𝛼, but if 𝛼 is an ordinal, then 𝐴 must be a thin set. If, however, 𝐴 is restricted to be a subset of the powers of 2, then not every order type is possible, although arbitrarily large countable well orders arise.