# Lattice initial segments of the hyperdegrees

Lattice initial segments of the hyperdegrees
(with Richard A. Shore). Journal of Symbolic Logic 75 (2010), no. 1, 103—130.

I was a postdoc with Shore during the academic year 2006-2007 at Cornell, and this paper was the result of our joint work there. It concerns hyperarithmetical reducibility and its induced partial ordering of the reals.

# Higher Kurtz randomness

Higher Kurtz randomness
(with André Nies, Frank Stephan, and Liang Yu). Annals of Pure and Applied Logic 161 (2010), no. 10, 1280—1290.

I visited Liang Yu at Nanjing University, China in 2008 and 2009. The collaboration for this paper however was later done by email plus a strategy meeting in Marseille in 2009.

# The probability distribution as a computational resource for randomness testing

The probability distribution as a computational resource for randomness testing
Journal of Logic and Analysis 2 (2010), no. 10, 1—13.

This paper grew out of my assignment to teach Math 472, Statistical Inference, in Spring 2009 at UH. By analyzing the proof of the law of large numbers and some other work I showed that Hippocratic and Galenic randomness coincide for Bernoulli measures. [There has been follow-up work to this paper which it would make sense to describe here.]

# Effective dimension of points visited by Brownian motion

Effective dimension of points visited by Brownian motion
(with Anil Nerode). Theoretical Computer Science 410 (2009), no. 4-5, 347—354.

Preliminary version: The law of the iterated logarithm for algorithmically random paths of Brownian motion.
Logical Foundations of Computer Science, Lecture Notes in Computer Science, vol. 4514, Springer, Berlin, 2007, pp. 310—317.

In this paper we answered a question of Fouché regarding algorithmically random Brownian motion. The key idea was to use Carathéodory’s measure algebra isomorphism theorem.

# Infinite subsets of random sets of integers

Infinite subsets of random sets of integers
Mathematical Research Letters 16 (2009), no. 1, 103—110.

This paper was written in Fall 2007 in Hawaii. It was superseded by a paper in J. Mathematical Logic 2011.

# Finding paths through narrow and wide trees

#### History of the paper

###### 2006

The paper was written at UConn.

###### 2009

Paper appeared in print:

Finding paths through narrow and wide trees
(with Stephen E. Binns). Journal of Symbolic Logic 74 (2009), no. 1, 349—360.

###### 2012

Laurent Bienvenu and Paul Shafer discovered an apparent error in the proof of Lemma 4.6. There it is stated that A wtt-reduces to B; however, it seems that the reduction of A to B also requires oracle access to f. Corollary 4.7 also seems false (take f to be a ML random of hyperimmune-free degree, and A=f; then A is complex but not f-complex).

# Self-embeddings of computable trees

Self-embeddings of computable trees (with Stephen Binns, Manuel Lerman, Jim Schmerl, and Reed Solomon).
Notre Dame Journal of Formal Logic 49 (2008), no. 1, 1—37.

I was probably the least-involved of the five authors of this paper, which was written in 2005-2006 at UConn.

# The strength of some combinatorial principles related to Ramsey’s theorem

The strength of some combinatorial principles related to Ramsey’s theorem
(with Denis R. Hirschfeldt, Carl G. Jockusch, Steffen Lempp, and Theodore A. Slaman.) Computational Prospects of Infinity. Part II: Presented Talks, Lecture Notes Series, Institute of Mathematical Sciences, National University of Singapore, vol. 15, World Scientific Publishing, Hackensack, NJ, 2008, pp. 143—162.

This paper established that Stable Ramsey’s Theorem for pairs implies DNR, which is a weak form of WKL.
It was later shown by Liu in 2011 that Ramsey’s theorem for pairs does not imply WKL (which is what we really wanted to know!).

This paper was written while all authors were participating in the Computational Prospects of Infinity workshop in Singapore in 2005.