"William Elliot" <marsh@[EMAIL PROTECTED]
> wrote in message
news:Pine.BSI.4.58.0804172334180.10208@[EMAIL PROTECTED]
> 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?
~~~~~~~~~~~~~~~~~~~~~~
I should not have used the word "probe"
if was going to throw you off that much.
However, the problem didn't originate in mathematics.
Moreover, in origin, it was probably closer in spirit to projective
geometry
(where the usual expression is "line on a point")
than to high school analytical geometry
(where I suppose they do speak of "line through a point").
It's been decades since I was in school,
and I'm not used to picayune criticism.
~~~~~~~~~~~~~~~~~~~~~~
> 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 ?
If you study math awhille longer
then you will encounter expressions such as
"by an abuse of notation ... - in order to avoid pedantry".
Also, I wouldn't be so sure as you are
that ZxZ is, strictly speaking, a "subset" of RxR.
But I know what you mean.
And I know (or presume that) you mean power-set by P(R),
although I have never seen that notation for it before.
Likewise, the shorthand notation
(A,B) =(by definition)= { (a,b) e S : a e A & b e B}
does occur. Whether you've ever seen it before or not.
And in any case you evidently didn't have any trouble
understanding what I meant.
~~~~~~~~~~~~~~~~~~~~~~
>> 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.)
>>
> What does a particular probe line mean?
>
> Do you mean, a given line will miss all the lattice
> points Z^2, iff it's has an irrational slope?
>
> That is false. Do you mean, a given line through the origin will
> miss all the lattice points ZxZ iff it's has an irrational slope?
>
> BTW, horizontal lines have rational slope
> and vertical lines require a special case,
> but rest assured, their slope isn't rational.
~~~~~~~~~~~~~~~~~~~~~~
All you're saying there
is obviously what I meant by - -
>> (excluding the horizontal and vertical lines,
>> and the point at the origin, in all this.)
~~~~~~~~~~~~~~~~~~~~~~
>> Another way to say it is that "almost all" probe lines
>> miss all the lattice points.
>>
>> 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.
It implicitly defined what I mean by a "probe line" in everything that
followed.
"probe line"
means
"a line on (or through) the origin".
~~~~~~~~~~~~~~~~~~~~~~
>> And it's not hard to see that, given any particular slope w, then disks
>> of dimini****ng radii, (each radius depending on the particular lattice
>> point), can be constructed so that this particular probe line misses
all
>> the disks.
~~~~~~~~~~~~~~~~~~~~~~
Well, that WAS a real MISTAKE on my part.
I meant, obviously, "given any particular irrational slope w".
~~~~~~~~~~~~~~~~~~~~~~
>>
> False, let w = 1/2.
>
>> And then it's immediate that you can select any finite or countable set
>> 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".
However, since, (as you point out with w=1/2)
the assertion is blatantly false as stated,
I should think it ought to have occured to you
that I must have meant irrational slopes.
Now, assuming that you understand that now,
if, then, still, you think the assertion is false,
then you just aren't thinking hard enough.
The distance of the probe line of slope w
to the lattice point (i,j) is
R(w,i,j,)
= | w - j/i | ( i / sqr(1+w²) )
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) = 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.
And then, using proof by leap of faith,
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.
Thank you for trying to help anyway,
~greg.
~~~~~~~~~~~~~~~~~~~~~~~~~
>> It seemed to me, at first, that there might be a way
>> to construct dimini****ng radii so that, once again,
>> almost-all probe lines miss all the disks.
>
> Let L = { 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 = 1/2
>
> No matter how small of a disk you put around (1,1)
> infinitely many of L will pass through the disk.
>
> Thus the proposition you stated to which I said "Again false"
> is still false with the added hypothesis that all the slopes
> are irrational.
>
>> But now I think that's not be the case.
>> But I don't know.
>>
>> So my question is, -- am I right that --that's not the case?
>
> No you're not right for sloppy statement of your propositions
> and you're not right for the corrected version of the
> proposition discussed above.
>
>> Is there is really no way to construct disks on lattice points
>> such that almost-all probe lines miss all the disks?
>
> Only if the set of lines is finite and all have irrational slope.
>
> ----
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