- 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})
`@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.