ideal delta-sigma encoding

A column from an '01 or '02 issue of American Scientist
claimed that the optimal number system is base 3, as far
as minimizing radix * the average number of digits in
numbers goes. This was shown empirically, but a passing
mention was made to the fact that 3 is the nearest
integer to e. Anybody has any idea how e fits in here
I'd be interested to hear about it. In any case it
suggests to me an audio encoder with base-3 delta-sigma
sampling.

But what happens if we permit unequally-sized bins.
Gene's recent Perfect Difference Set thing led me to...

http://mathworld.wolfram.com/PerfectRuler.html

Comments?

-Carl

> A column from an '01 or '02 issue of American Scientist
> claimed that the optimal number system is base 3, as far
> as minimizing radix * the average number of digits in
> numbers goes. This was shown empirically, but a passing
> mention was made to the fact that 3 is the nearest
> integer to e. Anybody has any idea how e fits in here
> I'd be interested to hear about it. In any case it
> suggests to me an audio encoder with base-3 delta-sigma
> sampling.
>
> But what happens if we permit unequally-sized bins.
> Gene's recent Perfect Difference Set thing led me to...
>
> http://mathworld.wolfram.com/PerfectRuler.html
>
> Comments?

The encoding, perfect ruler, and ideal coinage problems
look the same to me...

http://www.cs.uwaterloo.ca/~shallit/Papers/change2.pdf

-Carl