On Tue, Aug 13, 2013 at 2:31 AM, Henry Baker <hbaker1@pipeline.com> wrote:
It occurred to me that this is a classical problem of factoring with Gaussian integers.
The best 'denominators' are the most composite of Gaussian integers, so that you have the widest variety of angles represented.
In particular, pick denominators which are the product of 2 with all of the (rational integer) primes of the form 4n+1.
Turn on a light at the origin of the complex plane, put up thin barriers at every Gaussian integral point, and distribute some photograhic film around the circumference of a circle of radius r. Wherever the light shines through the most, those are angles that aren't well represented. In particular, I'm interested in the largest _gap_ in this circle for a given radius r; the size of that gap indicates the worst case for that radius r.
I did a few things close to what you describe. It can be downloaded here http://list.seqfan.eu/extfiles/Gaussian%20Shadow.pdf I draw a thin blue line through every Gaussian integer (here with absolute value less than 120) and I display a corona between two concentric circle (here 50 and 35) so that one can appreciate visually the resulting density. http://list.seqfan.eu/extfiles/Gaussian%20Prime%20Shadow.pdf is the same idea, restricted to Gaussian primes. The angles less represented are the ones close to exact rational approximation, it reminds of the noman's land around a potent bacterial colony. You find similar isolation around algebraic integers with small coefficients. Olivier