Kees Height and Proper/Improper Height Functions

When determining whether or not a function is a height function, we usually test it to see if it meets the following criteria:

1. Given any constant C, there are finitely many elements q such that H(q) <= C.
2. There exists a constant K such that H(q) >= K, for all q.
3. H(q) = H(1/q).
4. H(q^n) >= H(q) for any non-negative integer n.

Where q here represents any positive rational.

Some of these are arbitrary, but the first restriction is the ultimate dealbreaker - if a function doesn't bound a finite number of ratios under any arbitrary constant, it isn't a height.

This brings me to Kees Height:

Consider the sequence 1/1, 2/1, 4/1, 8/1 … 2^n/1. All of these have a Kees Height of 1, and therefore Kees Height breaks rule #1 of height functions.

However, if we apply an equivalence relation on the rationals (such as octave-equivalence), we can potentially solve the problem, and give way to a modified height function. I propose to call these heights Improper Heights, since they do not meet the standards without an equivalence relation on the rationals.

I've already edited the wiki to include this example. Is "improper" a good descriptor? Or is there an even better name I could use? (e.g. "non-strict height," or "semi-height," etc).

Ryan Avella

On Mon, Sep 10, 2012 at 2:05 AM, domeofatonement <domeofatonement@yahoo.com>
wrote:
>
> I've already edited the wiki to include this example. Is "improper" a good
> descriptor? Or is there an even better name I could use? (e.g. "non-strict
> height," or "semi-height," etc).

I like "semiheight." The Weil norm is a norm, and Kees expressibility
is a seminorm. Thus, if Weil norm is also a height, Kees
expressibility can be a semiheight.

I still wonder why you like criterion 2 as its written, rather than
the stronger criterion that H(q) must be nonnegative. What's the use
of allowing the lower bound on a height function to be less than 0?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> I still wonder why you like criterion 2 as its written, rather than
> the stronger criterion that H(q) must be nonnegative. What's the use
> of allowing the lower bound on a height function to be less than 0?
>
> -Mike
>

Maybe Gene can comment on this. I remember him saying before that he doesn't like it when mathematicians put arbitrary restrictions on heights, so I tried to leave it open for unforseen boundary cases.

Ryan

--- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > I still wonder why you like criterion 2 as its written, rather than
> > the stronger criterion that H(q) must be nonnegative. What's the use
> > of allowing the lower bound on a height function to be less than 0?
> >
> > -Mike
> >
>
> Maybe Gene can comment on this. I remember him saying before that he doesn't like it when mathematicians put arbitrary restrictions on heights, so I tried to leave it open for unforseen boundary cases.

I can't recall any negative heights. But H(q)>=H(1) seems fine.