Representing Ordinal Numbers with Arithmetically Interesting Sets of Real Numbers

Abstract

For a real number $x$ and set of natural numbers $A$, define $x \ast A := \{ x a \bmod 1: a\in A\}\subseteq [0,1).$ 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 a well order, 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.