Consider the lattice of integral points (i,j) in (Z,Z) in (R,R),
and probe lines on the origin, shooting through the lattice.
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.)
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.
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.
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.
~~
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.
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?
Is there is really no way to construct disks on lattice points
such that almost-all probe lines miss all the disks?
~greg
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