Representing Ordinal Numbers with Arithmetically Interesting Sets of\n Real Numbers

For a real number $x$ and set of natural numbers $A$, define $x \\ast A := \\{\nx a \\bmod 1: a\\in A\\}\\subseteq [0,1).$ We consider relationships between $x$,\n$A$, and the order-type of $x\\ast A$. For example, for every irrational $x$ and\norder-type $\\alpha$, there is an $A$ with $x\\ast A \\simeq \\alpha$, but if\n$\\alpha$ is a well order, then $A$ must be a thin set. If, however, $A$ is\nrestricted to be a subset of the powers of 2, then not every order type is\npossible, although arbitrarily large countable well orders arise.\n