Calkin-Wilf vs Benedetti

On Wed, Jan 2, 2013 at 10:35 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> for each finitely generated group G of positive rationals, and particular
> for any p-limit, we can form the sum
>
> dense(G) = sum_{q in G} 1/calkinv(q)
>
> which will converge. What the heck does this tell us?

It also looks like it converges if you use Benedetti Height instead. I
have a formula for "nice" subgroups, where a "nice" subgroup is one
where the T1 norm is also just a distorted L1 norm.

To prove this, first note that the sum of the reciprocal of Benedetti
height for every interval on the prime p axis is 2*sum_n (1/p)^n for
n=0..Inf, which by the usual formula for the convergence of a
geometric series is 2*1/(1-(1/p))=2p/(p-1). Then note that for reduced
fractions with no prime factors in common, H(a) * H(b) = H(ab), where
H is Benedetti height. Thus, for two primes p and q, the sum of the
reciprocal of Benedetti height for every interval on the subgroup
generated by {p,q} can be factored as (2*sum_n (1/p)^n)*(2*sum_n
(1/q)^n), which can then be rewritten as 2^2 * (p/(p-1)) * (q/(q-1)).

The same principle can then be extended to finitely-generated
subgroups of any rank, with the general expression for some rank-r
"nice" subgroup m_1.m_2.m_3...m_n being 2^r * prod_k(m_k)/(m_k-1) for
k=1..n.

Here are some examples:
2.3.5: 30
4.3.5: 20
2.9.5: 22.5
2.3.25: 25
4.9.25: 12.5

You can see that lower numbers = more complex. Furthermore, since
4.3.5 and 2.9.5 don't agree, we can be certain that this is no simple
function on multimonzos.

What we're really doing is adding 1/(2^||v||), for all v in some
lattice in interval space, where ||v|| is the T1 norm. Here are some
things I'm very curious about:
1) How to induce some natural and similar measure for lattices in the dual space
2) Using temperamental complexity, which is just the quotient of the
T1 norm mod some subspace of unison vectors, to do the same thing for
tempered spaces and come up with a measure of temperament complexity
3) How this works for non-nice subgroups
4) How all of these infinite sums relate to the zeta function

And some other stuff too.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia wrote:

> It also looks like it converges if you use Benedetti Height instead.

Good plan. I just used CalkinWilf because it was on my mind to find a use for it.

--- In tuning-math@yahoogroups.com, Mike Battaglia wrote:

Then note that for reduced
> fractions with no prime factors in common, H(a) * H(b) = H(ab), where
> H is Benedetti height.

This is also true for reduced fractions with no nonzero powers of a basis element in common. Every element of the subgroup has a unique factorization into powers of basis elements, this is not just true for primes or prime powers.

--- In tuning-math@yahoogroups.com, "genewardsmith" wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
>
> Then note that for reduced
> > fractions with no prime factors in common, H(a) * H(b) = H(ab), where
> > H is Benedetti height.
>
> This is also true for reduced fractions with no nonzero powers of a basis element in common. Every element of the subgroup has a unique factorization into powers of basis elements, this is not just true for primes or prime powers.
>

So the question is, how can we get this unique factorization to actually work? If we have a subgroup 2.3.7/5, then everything in it can be written in the form 2^a 3^b (7/5)^c. The Benedetti height of that is 2^|a| 3^|b| 35^|c|. What about something like 2.9/5.9/7? The height of 2^a (9/5)^b (9/7)^c is 2^|a| 9^(|b|+|c|) 5^|b| 7^|c| = 2^|a| 45^|b| 63^|c|. Food for thought.

On Thu, Jan 3, 2013 at 12:01 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> This is also true for reduced fractions with no nonzero powers of a
> basis element in common. Every element of the subgroup has a unique
> factorization into powers of basis elements, this is not just true for
> primes or prime powers.

Yes, but H(a)*H(b) = H(ab) only holds true for nice subgroups.

The thing I wrote about "primes" was too specific and a hangover from
an earlier edit.

> So the question is, how can we get this unique factorization to actually
> work? If we have a subgroup 2.3.7/5, then everything in it can be written in
> the form 2^a 3^b (7/5)^c. The Benedetti height of that is 2^|a| 3^|b|
> 35^|c|. What about something like 2.9/5.9/7? The height of 2^a (9/5)^b
> (9/7)^c is 2^|a| 9^(|b|+|c|) 5^|b| 7^|c| = 2^|a| 45^|b| 63^|c|. Food for
> thought.

I don't know, but I expect the answer will end up using V-maps again
in some way, which I still wish I had named subgroup matrices.

-Mike

On Thu, Jan 3, 2013 at 4:40 AM, Mike Battaglia <battaglia01@gmail.com>
wrote:
>
> Here are some examples:
> 2.3.5: 30
> 4.3.5: 20
> 2.9.5: 22.5
> 2.3.25: 25
> 4.9.25: 12.5
>
> You can see that lower numbers = more complex. Furthermore, since
> 4.3.5 and 2.9.5 don't agree, we can be certain that this is no simple
> function on multimonzos.

Here's a good question: is there any variant of this technique which
will yield a function with some theoretically nice properties? For
instance, it would be really nice a function existed which said that
4.9.25 was twice as complex as 2.3.5, which would make this function
homogeneous of degree 1. We're just doing Sum 1/benedetti height, but
is there a related and nicer way to do things?

-Mike