On Fri, 18 Apr 2008, ~greg wrote:
> "William Elliot" <marsh@[EMAIL PROTECTED]
> wrote
> > On Thu, 17 Apr 2008, ~greg wrote:
> >
> >> Consider the lattice of integral points (i,j) in (Z,Z) in (R,R),
> >> and probe lines on the origin, shooting through the lattice.
> >>
> > What is a probe line on the origin?
> > Do you mean a line through the origin?
> It's been decades since I was in school,
> and I'm not used to picayune criticism.
>
Math is exacting, so get used to it.
> > The set of lattice points is ZzZ and the real plane
> > is RxR. (Z,Z) and (R,R) are ordered pairs of two
> > elements from P(R).
>
> ZzZ ?
>
AxB =3D { (a,b) | a in A, b in B }, the set of all ordered pairs,
with the first element from A and the second element from B,
is the cross product of A and B
> Also, I wouldn't be so sure as you are
> that ZxZ is, strictly speaking, a "subset" of RxR.
>
If (n,m) in ZxZ, then n,m in Z, hence n,m in R and thus (n,m) in RxR.
> And I know (or presume that) you mean power-set by P(R),
> although I have never seen that notation for it before.
>
Yes.
> And in any case you evidently didn't have any trouble
> understanding what I meant.
>
The word probe threw me off, requring deciphering of what you meant and
your ommision lines through the origin and then again the ommision of
irrational slope, left you not stating what you meant. A person with less
patience or less ability, wouldn't understand.
> >> Then it is easy to see that a particular probe line
> >> will miss all the lattice points IFF its slope is irrational
> >> (excluding the horizontal and vertical lines,
> >> and the point at the origin, in all this.)
> >> And it's also easy to see that if disks of a fixed finite radius
> >> are constructed on each lattice point, then no probe line
> >> will miss all of them.
>
> > Again, are you using probe line to mean line through the origin? Hm,
> > that's so because every real number is arbitrarily close to a
rational.
>
> Good. You got that.
>
> But again, I started this whole thing off by saying
>
> >> Consider the lattice of integral points (i,j) in
> >> (Z,Z) in (R,R), and probe lines on the origin,
>
> That limited the universe of discourse.
>
Ok.
> >> And it's not hard to see that, given any particular slope w, then
disk=
s
> >> of dimini****ng radii, (each radius depending on the particular
lattice
> >> point), can be constructed so that this particular probe line misses
a=
ll
> >> the disks.
>
> Well, that WAS a real MISTAKE on my part.
>
> I meant, obviously, "given any particular irrational slope w".
>
> > False, let w =3D 1/2.
> >
> >> And then it's immediate that you can select any finite or countable
se=
t
> >> of probe lines and construct disks of dimini****ng radii such that
none
> >> of these particular lines hit any of the disks.
> >>
> > Again false.
>
> Again, I meant, obviously,
> "any finite or countable set of probe lines of irrational slope".
>
It wasn't obvious. It had to be deduced.
> However, since, (as you point out with w=3D1/2)
> the assertion is blatantly false as stated,
> I should think it ought to have occured to you
> that I must have meant irrational slopes.
>
It should occure to you to proof read to assure completenss and accuracy
is what you assert. Math is an exacting science.
> The distance of the probe line of slope w
> to the lattice point (i,j) is
>
> R(w,i,j,)
> =3D | w - j/i | ( i / sqr(1+w=B2) )
>
> So, all you have to do is make each disk on each (i,j)
> have a smaller radius than that.
>
> Or better, just use the interior of the disks of radii Rij.
>
> Then the probe-line w will miss all the disks
> on all lattice points.
>
> And then using R(W,i,j) =3D min { R(w,i,j) : w e W }
> you get the same assertion for a given finite set W
> of irrational (of course) slopes w.
>
> Now, using proof by waving of hands,
> I asserted (I think I said "it's easy" or "obvious")
> that the same thing holds for countable sets W.
>
It does not because the minimum may be zero.
I gave a detailed counter example that showed just than.
> And then, using proof by leap of faith,
Math is not religion. That which is apparent for all finite beings, holds
not for infinite beings. For example for all n in N,
=090 < 1/n
and
=09inf{ 1/n | n in N } =3D 0
If you don't understand infinums, consider inf like min.
> I thought, at first, that the same might hold
> for particular uncountable sets W too,
> although obvioulsy not for all.
>
> But then I thought that might not be the case.
> Which is the question I asked.
> The rest of your response here is stuck on my failure to have explicitly
> stated that I meant irrational w.
>
Read my counter example, where I explicity stated that
all the slopes s_j were irrational.
> Thank you for trying to help anyway,
>
I've given you a counter example. Now read it.
> > Let L =3D { L_j | j in N } be a countable set of lines through the
> > origin (0,0), of the real plane R^2. For all j in N, let s_j be the
> > slope of L_j. Construct L so that for all j in N, s_j is irrational
> > and
> >
> > lim(j->oo) s_j =3D 1/2
> >
> > No matter how small of a disk you put around (1,1)
> > infinitely many of L will pass through the disk.
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