Spectral Theory of Regular Sequences

Explore the spectral theory of regular sequences in this 55-minute conference talk by Michael Coons from the University of Bielefeld. Delve into k-regular sequences, Stern's sequence, and the complexity of integer sequences. Examine the finiteness conjecture for maximal values and investigate the Zaremba sequence. Learn about ghost measures and their relation to dilation equations and iterated function systems. Discover the IFS and ghost distribution of Zaremba's sequence, and understand how to determine spectral types. Investigate continuous ghost measures and their level-set construction, with examples from the Stern sequence and 2-Zaremba sequence. Conclude by considering open questions and potential areas for further research in this field of mathematics.

Intro
k-regular sequences
Stern's sequence
How complicated are integer sequences?
Three questions...
Maximal values: the finiteness conjecture
The Zaremba sequence
General existence of ghost measures, I
Proof via relation to dilation equations
From dilation equations to iterated function systems
The IFS and ghost distribution of Zaremba's sequence
Determining the spectral type
Continuous ghost measures have a level-set construction
Example: Stern sequence
Examples: 2-Zaremba sequence
Some questions and further work