• D. D. Blair, J. D. Hamkins, and K. O’Bryant, “Representing Ordinal Numbers with Arithmetically Interesting Sets of Real Numbers,” Integers, vol. 20A, 2020. (Paper~A3, \url{http://math.colgate.edu/~integers/vol20a.html})  
  Citation arχiv

  @ARTICLE{BlairHamkinsOBryant2020:Representing-ordinal-numbers-with-arithmetically-interesting-sets-of-real-numbers,
  author = {D. Dakota Blair and Joel David Hamkins and Kevin O'Bryant},
  title = {Representing Ordinal Numbers with Arithmetically Interesting Sets of Real Numbers},
  journal = {Integers},
  FJOURNAL = {Integers Electronic Journal of Combinatorial Number Theory},
  year = {2020},
  volume = {20A},
  number = {},
  pages = {},
  month = {},
  note = {Paper~A3, \url{http://math.colgate.edu/~integers/vol20a.html}},
  abstract = {},
  keywords = {},
  source = {},
  doi = {},
  url = {https://wp.me/p5M0LV-1Tg},
  eprint = {1905.13123},
  archivePrefix = {arXiv},
  primaryClass = {math.NT},
  }

Abstract. For a real number $x$ and set of natural numbers $A$, define $x∗A = \{xa \mod 1 \mid a \in A\}\subseteq [0,1)$. We consider relationships between $x$, $A$, and the order-type of $x∗A$. For example, for every irrational $x$ and order-type $\alpha$, there is an $A$ with $x ∗ A \simeq\alpha$, 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.