[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 $x\ast A=\{xa\mathrm{mod}1\mid a\in A\}\subseteq [0,1)$. We consider relationships between $x$, $A$, and the order-type of $x\ast A$. For example, for every irrational $x$ and order-type $\alpha$, there is an $A$ with $x\ast A\simeq \alpha$, but if $\alpha$ 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.