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?
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).
> 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.
> 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.
> 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.
>
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.
> 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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