Justin D. Miller

I am a John Wesley Young postdoc at Dartmouth College. I did my undergrad at the Pennsylvania State University, where Jason Rute supervised my undergraduate thesis. I earned my PhD from the University of Notre Dame under the supervision of Peter Cholak. My area of research is mathematical logic and computability theory, specifically asymptotic computability, algorithmic randomness, and reverse mathematics.

My CV is available here.

I can be contacted via email and my office is 315 Kemeny Hall.

Teaching

• During Winter Term 2022, I am teachin Math 13.
• During Fall Term 2021, I am teaching Math 13.

Research

Here is a list of my publications, along with preprints where available. The status of some resolved open questions can be found here.

• Intermediate intrinsic density and randomness. Computability, To appear. doi: 10.3233/COM-210309. (Preprint)
• Intrinsic Smallness. The Journal of Symbolic Logic, To appear. doi: 10.1017/jsl.2020.39. (Preprint)

My PhD dissertation is also available here.

Talks

The following is a list of the talks I have given at external meetings in reverse chronological order.

Status of Some Open Questions

This is a list of solutions to open questions appearing in my previous work, along with links and attributions where necessary.

Last Updated: 7/29/2021

Intrinsic Smallness

• Previously Question 4.2: There is an arithmetically intrinsically small set which is not $$\emptyset^{(\omega)}$$-intrinsically small. In fact, there is an $$\emptyset^{(\omega)}$$-computable set with this property, as $$\emptyset^{(\omega)}$$ can compute $$\emptyset^{(n+2)}$$ uniformly in $$n$$ and therefore can compute all arithmetical permutations.
• Question 4.1: For all $$X$$, the degrees of $$X$$-intrinsically small sets are exactly the $$X$$-high or $$X$$-DNC degrees. (This is Corollary 2.27 in my thesis.)

Dissertation

• Previously Question 4.2.6: The degrees of intrinsic density $$r$$ sets cannot be the high or DNC degrees because each degree contains only countably many sets, but there are uncountably many reals $$r$$. This was observed by Denis Hirschfeldt.