Diophantine approximation alternatives

Dave was questioning the lack of alternatives, so let's look at the
standard Diophantine approximation ones. In the 7-limit, the
n^(-1/3) exponent puts the solutions into a cube of side n^(-1/3),
and hence of volume 1/n. This gives a density of solutions
proportional to 1/n, and since the integral of 1/n is unbounded, an
infinity of solutions may be expected.

In general, if f(n)>0 is such that its integral is unbounded, then
for d irrational numbers xi,
max f(n)^(-1/d) |round(n*xi) - n*xi| < c
"almost always" has an infinite number of solutions. This isn't as
tight a theorem as when we use exponents, but in practice it works
for our problem.

Most obviously, we could use an exponent less than 1/3, so that using
fourth roots instead will still give an infinite number of soutions--
a fact which already is obvious without the above. We could even use
1/ln(x), and get solutions in droves, like prime numbers. On the
other hand, we could fade just a little faster by using 1/(n ln n),
which makes the high end die out more, but probably not quite enough
to kill off an infinite list of solutions.

I don't see any advantages here, but there it is.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Dave was questioning the lack of alternatives, so let's look at the
> standard Diophantine approximation ones.

Why not look outside Diophantine approximation alternatives?

> In general, if f(n)>0 is such that its integral is unbounded, then
> for d irrational numbers xi,
> max f(n)^(-1/d) |round(n*xi) - n*xi| < c
> "almost always" has an infinite number of solutions. This isn't as
> tight a theorem as when we use exponents, but in practice it works
> for our problem.
...
> I don't see any advantages here, but there it is.

There probably aren't any advantages here.

But why does badness have to be of the form
f(n)* |round(n*xi) - n*xi|
at all?

Why not
f(n) + |round(n*xi) - n*xi|
or
f(n) * g(|round(n*xi) - n*xi|)
?