Addenda
Spectral Proof of Theorem 7
We present an alternate proof of Theorem 7 that makes use of spectral theory and eigenvalue criteria rather than measure theory. This proof appeared in an earlier version of the paper but was replaced at the request of the referee.
Further Details on the Proof of Theorem 7
In response to questions regarding Theorem 7 we provide additional details on how the inequality is derived. Thanks in part to Alexandre Danilenko for pointing out these difficulties.
Further Details on Mixing Height Sequences
The crux of the paper is the argument relating mixing limits to ergodic average limits. We expound on the proof of mixing height sequences to help clarify the argument allowing the extraction of the 1/r term.
Restricted Growth on Polynomial Staircase Transformations
Here we outline the restricted growth condition specifically for polynomial staircase transformations and achieve a condition that is similar to that used by Adams in the original proof of the staircase being mixing. Thanks to Alexandre Danilenko for asking us to write this condition in a more explicit form.
Generalized Staircase Transformations
The class of generalized staircase transformations was found in Darren Creutz's Honors Thesis. We present here the class of such transformations and prove that they are also mixing transformations to provide a fuller set of examples of rank-one mixing.
Staircase Odometer Hybrid Transformations
The class of staircase odometer hybrid transformations, while not mixing, have some interesting properties. We present the class and apply our methods to studying the behavior of their height sequences.