The measure theoretic approach to the density of quasi-arithmetic sequences

A 20-minute talk at the 1999 Seaway Number Theory Conference, March 27-28, 1999, at Pennsylvania State University in State College, PA

In this talk I posed and gave evidence for a conjecture concerning a geometric inclusion/exclusion interpretation of integer shade densities of quasi-arithmetic sequences. The process evidently produces familiar sequences of continued fractions but I haven't been able to prove that. Although nobody at the conference was aware of it, it turns out that the main question had been answered already, albeit in a completely different (and non-constructive) way, in the paper

Richard Bumby and Erik Ellentuck, "Finitely additive measures and the first digit problem", Fund. Math., 65 (1969), 33-42.

The solution of Bumby and Ellentuck uses ultrafilters and is essentially an existence result. My idea is perhaps unusual in that it applies a different existence theorem (the Hahn-Banach theorem) to a constructive process. (One might well ask, What is the good of having a constructive process if you still need a non-constructive result on its heels? But I don't. Ask, that is.)

A PDF version of the slides of the talk is linked below, but doesn't quite catch the flavor of the presentation, which was pretty entertaining, if I do say so myself. It had to be, as I was an imposter in the midst of actual number theorists, begging for their help.

seaway99.pdf