Simplified, bite-sized version of the last question

Everything will be great if I can just figure out the answer to the
following question:

For a Farey series, each rational supposedly has a width of 1/d.
For a Tenney series, each rational supposedly has a width of 1/sqrt(n*d).
For the Stern-Brocot tree, each rational has a width of something else.

The question: for the full set of rationals, what is the width that
each rational has? Are there actually more rationals near 135/134 than
near 2/1?

-Mike

Mike wrote:

>The question: for the full set of rationals, what is the width that
>each rational has?

Zero.

>Are there actually more rationals near 135/134 than near 2/1?

Yes.

-Carl

On Mon, Jan 24, 2011 at 8:20 PM, Carl Lumma <carl@lumma.org> wrote:
>
> Mike wrote:
>
> >The question: for the full set of rationals, what is the width that
> >each rational has?
>
> Zero.

The question: for the full set of rationals, what is the width that
each rational has, divided by the total amount of rationals?

> >Are there actually more rationals near 135/134 than near 2/1?
>
> Yes.

The Farey series models this as there being a density of ~d, the
Tenney series ~sqrt(n*d). Things are a lot different for the Stern
Brocot tree - if you try to seed the entropy calculation with it, you
get a crazy looking curve. Since the two series converge as N goes to
infinity, it stands to reason that the densities would converge as
well. Do they converge on some well-defined value?

-Mike

On Mon, Jan 24, 2011 at 8:25 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
> On Mon, Jan 24, 2011 at 8:20 PM, Carl Lumma <carl@lumma.org> wrote:
>>
>> Mike wrote:
>>
>> >The question: for the full set of rationals, what is the width that
>> >each rational has?
>>
>> Zero.
>
> The question: for the full set of rationals, what is the width that
> each rational has, divided by the total amount of rationals?

Times* the total amount of rationals

-Mike

>> >The question: for the full set of rationals, what is the width that
>> >each rational has?
>>
>> Zero.
>
>The question: for the full set of rationals, what is the width that
>each rational has, divided by the total amount of rationals?

Zero.

>> >Are there actually more rationals near 135/134 than near 2/1?
>>
>> Yes.
>
>The Farey series models this as there being a density of ~d,

The mediant-mediant widths are proportional to d in a given
farey series, which will contains rationals of vastly different d.

>the Tenney series ~sqrt(n*d). Things are a lot different for the
>Stern Brocot tree - if you try to seed the entropy calculation
>with it, you get a crazy looking curve. Since the two series
>converge as N goes to infinity, it stands to reason that the
>densities would converge as well. Do they converge on some
>well-defined value?

What converges and how?

-Carl

On Mon, Jan 24, 2011 at 8:37 PM, Carl Lumma <carl@lumma.org> wrote:
>
> >> >The question: for the full set of rationals, what is the width that
> >> >each rational has?
> >>
> >> Zero.
> >
> >The question: for the full set of rationals, what is the width that
> >each rational has, divided by the total amount of rationals?
>
> Zero.

I meant times, see the correction.

> >> >Are there actually more rationals near 135/134 than near 2/1?
> >>
> >> Yes.
> >
> >The Farey series models this as there being a density of ~d,
>
> The mediant-mediant widths are proportional to d in a given
> farey series, which will contains rationals of vastly different d.

Right.

> >the Tenney series ~sqrt(n*d). Things are a lot different for the
> >Stern Brocot tree - if you try to seed the entropy calculation
> >with it, you get a crazy looking curve. Since the two series
> >converge as N goes to infinity, it stands to reason that the
> >densities would converge as well. Do they converge on some
> >well-defined value?
>
> What converges and how?

Gene answered this in the other thread, and apparently my question is
equivalent or somehow related to the Riemann hypothesis. Now I'm
irritated that I asked, because if I had just kept working, maybe I'd
have solved it without realizing what I'd done! Now I'm all thrown
off, of course. Just kidding.

The premise is:
For any series, let the "relative width" of some rational a/b denote
its absolute width, taken by mediant-to-mediant, divided by the
absolute width of 1/1.

Now, the questions are:
1) As any series is left to run till infinity, will its relative width
approach some kind of simple function of a and b that holds true for
all a/b?
2) If so, will this function be the same for ALL series, or is it
dependent on the kind of series you use?

Or put more simply: the Farey series ratios have an approximate width
of 1/d. The Tenney series ratios have an approximate width of
1/sqrt(n*d). As N increases and these series incorporate more ratios,
do these approximations converge on some actual formula, holding true
for both series, of which 1/d and 1/sqrt(n*d) are just approximations?

-Mike

>Times* the total amount of rationals

Transfinite arithmetic is not my strong suit. Wolfram Alpha
gives zero for this, and undefined for your previous question,
for aleph-0.

-Carl

>The premise is:
>For any series, let the "relative width" of some rational a/b denote
>its absolute width, taken by mediant-to-mediant, divided by the
>absolute width of 1/1.

What is the mediant above 1/1?

>Now, the questions are:
>1) As any series is left to run till infinity, will its relative width
>approach some kind of simple function of a and b that holds true for
>all a/b?

The widths are all zero in the limit. At any finite point, I would
guess they retain their relative widths.

>2) If so, will this function be the same for ALL series, or is it
>dependent on the kind of series you use?

Any series that includes all the rationals should behave the
same way, obeying its relative widths rule to the limit.

>Or put more simply: the Farey series ratios have an approximate width
>of 1/d. The Tenney series ratios have an approximate width of
>1/sqrt(n*d). As N increases and these series incorporate more ratios,
>do these approximations converge on some actual formula, holding true
>for both series, of which 1/d and 1/sqrt(n*d) are just approximations?

The Tenney series isn't well-studied that I know of. What's
its extension rule? Add one to the Tenney limit? Do new ratios
appear in an orderly fashion as we extend with this rule?
Are we using mediants for widths?

-Carl

On Mon, Jan 24, 2011 at 8:57 PM, Carl Lumma <carl@lumma.org> wrote:
>
> >The premise is:
> >For any series, let the "relative width" of some rational a/b denote
> >its absolute width, taken by mediant-to-mediant, divided by the
> >absolute width of 1/1.
>
> What is the mediant above 1/1?

That depends on N...

The question is, what does the relative width of a/b change as N tends
to infinity?

> >Now, the questions are:
> >1) As any series is left to run till infinity, will its relative width
> >approach some kind of simple function of a and b that holds true for
> >all a/b?
>
> The widths are all zero in the limit. At any finite point, I would
> guess they retain their relative widths.

The absolute widths are zero in the limit, but are the relative widths?

I guess this is another statement equivalent to the Riemann
Hypothesis, because the regularity of the distribution of the
rationals is surely related to the regularity of the distribution of
the primes? Specifically the rational-indicator function can probably
be represented as the prime-indicator function convolved with itself
an infinite number of times, or perhaps convolved in log space, or
perhaps convolved in the Mellin sense.

What that last sentence means is that I don't know exactly how to do
it, but I'm sure you could formalize the relationship, and then come
up with an analogous Zeta function for the rational-counting function
rather than the prime-counting function. Since we like weighting the
rationals around here, then an analogous relationship might be able to
be drawn between the weighted prime-indicator function and the
weighted rational-indicator function.

And if that's the case then perhaps we could come up with a
rational-counting version of the Zeta function, assume the Riemann
Hypothesis, and then work with that from now on.

> >2) If so, will this function be the same for ALL series, or is it
> >dependent on the kind of series you use?
>
> Any series that includes all the rationals should behave the
> same way, obeying its relative widths rule to the limit.

But that of the Farey and Tenney series differ...

> >Or put more simply: the Farey series ratios have an approximate width
> >of 1/d. The Tenney series ratios have an approximate width of
> >1/sqrt(n*d). As N increases and these series incorporate more ratios,
> >do these approximations converge on some actual formula, holding true
> >for both series, of which 1/d and 1/sqrt(n*d) are just approximations?
>
> The Tenney series isn't well-studied that I know of. What's
> its extension rule? Add one to the Tenney limit? Do new ratios
> appear in an orderly fashion as we extend with this rule?
> Are we using mediants for widths?

These are really good questions. Yes, let's use mediants for widths. I
we could use means as well, if we want, but we might come up with a
different answer.

-Mike

>> What is the mediant above 1/1?
>
>That depends on N...

Check again.

>> The widths are all zero in the limit. At any finite point, I would
>> guess they retain their relative widths.
>
>The absolute widths are zero in the limit, but are the relative widths?

What's 0/0?

>> Any series that includes all the rationals should behave the
>> same way, obeying its relative widths rule to the limit.
>
>But that of the Farey and Tenney series differ...

Yes, I meant each obeying its own.

-Carl

On Mon, Jan 24, 2011 at 9:13 PM, Carl Lumma <carl@lumma.org> wrote:
>
> >> What is the mediant above 1/1?
> >
> >That depends on N...
>
> Check again.

OK, sorry, I see what's going on now. How was Paul using the Farey
series here, then? Was he just taking the reciprocal of the series to
seed HE?

I guess to really formalize this, we'd have to work with an extension
of the Farey series that goes between 0/1 and 1/0, rather than 0/1 and
1/1. The difference between this and the Stern-Brocot tree is that
entries from the tree would still be pruned if their denominators are
> N. Let's call this a w-Farey series for the moment, where w is
actually Omega.

Or maybe an analogous question is just to rephrase the question for
the Farey series, but use 1/2 as the relative point instead of 1/1,
and phrase it for every interval except the endpoints.

> >> The widths are all zero in the limit. At any finite point, I would
> >> guess they retain their relative widths.
> >
> >The absolute widths are zero in the limit, but are the relative widths?
>
> What's 0/0?

You'd have to work out the limit and see what it comes out to. I think
this is what Gene was getting at with the ?(x) function in the other
thread.

> >> Any series that includes all the rationals should behave the
> >> same way, obeying its relative widths rule to the limit.
> >
> >But that of the Farey and Tenney series differ...
>
> Yes, I meant each obeying its own.

But I think that what's going on is that, assuming the Riemann
hypothesis, they would converge. I think.

-Mike