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Martin Sleziak
Martin Sleziak
5:52 PM
Have you tried just to write a computer program to find the exceptional values? They are indeed, finitely many; as soon as the region contains a unit square in every direction (for which it is enough to contain a disk of diameter $\sqrt 2$), it has an integer point in it regardless of the orientation, so for all $b$ with $|b|>15$, say, all $4$ possibilities are realized and (taking symmetries into account), you need to check just about $100$ values, which should be a piece of cake. — fedja 22 hours ago
@fedja I have posted an answer showing the distribution for small enough numbers. I am not sure to which extent it is similar to what you intended.
I have to admit that I thought that there would be more exceptions. For some reason I kept thinking about integer multiples of $\frac1b$ and integer multiples of $\frac ib$; but somehow I ignored that also their sums come into play.
 
fedja
fedja
6:17 PM
The pictures are cool :-) What do you use for making them? (I usually stick with Asymptote for most of my drawings but I'm curious what the preferences of other people are in this respect)
 
Martin Sleziak
Martin Sleziak
I am pretty lame in drawing pictures - MetaPost is basically the only thing I am familiar with well enough to create some reasonable picture.
WP: MetaPost Here is some kind of manual: tug.org/docs/metapost/mpman.pdf
I'd guess several other tools are more popular:
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A: TeX Community Polls

Martin ScharrerI create the diagrams for my (La)TeX documents using ... Possible choices: TikZ, PGF, PSTricks, xypic, picture environment, MS Visio, Inkscape, ... Select 'PGF' if you really using the low level PGF commands, but 'TikZ' if you use that higher level interface.

From MetaPost I get the picture basically in ps (or eps) format. Which is probably good for use in LaTeX documents, but for use on Stack Exchange I have to convert them after that.
 

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