The generalization of Dan's question comes from the "Markov Spectrum" http://en.wikipedia.org/wiki/Markov_spectrum If we define an equivalence relation on reals by x is equivalent to (a*x + b)/(c*x+d) for all integer matrices (a,b;c,d) of determinant 1 (i.e. fractional linear transformations), you can see that two reals are equivalent if and only if after some finite time the sequence of convergents of their continued fractions agree. So Hurwitz proved that the worst approximable numbers are those equivalent to (sqrt(5)-1)/2. If you exclude those, you ask for the next worst approximable, and so on. Victor On Fri, Mar 23, 2012 at 4:10 PM, Warren Smith <warren.wds@gmail.com> wrote:
Re Asimov's conjecture, the fact that g=(sqrt(5)-1)/2 is the (essentially unique) "worst approximable" (by rationals) number is known and is rigorous and basically comes from fact its continued fraction is [1,1,1,1,1,...]
Indeed there is a known theorem by A.Hurwitz and/or E.Borel somewhere in 1890-1905 that among any 3 successive CF convergents p/q, at least one must be as close or closer than 1/(sqrt(5)*q^2)... and sqrt(5) is the best possible constant for rational approximations because the golden ratio g meets the bound...
This means the Weyl sequence 0, g, 2g, 3g, 4g... mod 1 has minimum gap shrinking asymptotically as slowly as possible for any Weyl sequence. But Asimov wanted the max gap to shrink as quickly as possible.
For any Weyl sequence n*g mod 1, n=0,1,2,3... in the case where g is any quadratic irrational (periodic CF) the max and min gaps must shrink at the same rate which is gaps proportional to 1/#points. Plainly the two gaps to the point "0" are the only ones that matter by translation invariance (which is enough-valid) which hint should be good enough for you to prove. So I claim for any Weyl sequence based on any quadratic irrational, the max and min gaps both shrink like constants times 1/#points, and those plainly are best possible up to the values of the constants.
Also this paper may be of interest (have not seen): N.B.Slater: Gaps and steps for the sequence nX mod 1, Proc. Cambridge Philos. Soc. 63 (1967) 1115-1123.
About Propp's question, what are those hyperbolas etc, the sunflower pattern is basically a point lattice, except you've distorted things by use of polar coordinates. You would actually have had a genuine lattice if you'd drawn it on the surface of a cylinder spacing points along a geodesic, i.e. helix, at constant spacing. Point lattices contain lines. Your sunflowers therefore contain distorted lines, i.e. curves. If you view cylinder as a cone of angle 0 and flat paper as a cone of angle 180, and cones as angles a with 0<a<180 then the map consists of increasing the cone angle from 0 to 180 thus mapping the cylinder-lattice-pattern onto the plane-sunflower-pattern. Obviously, conic curves seem likely to be important... This is not yet a full explanation, but I'm confident is tied to the correct explanation.
Now to return to my problem about K-colored generalization of sunflowers, how about some colored postscript code for them using multiple colors, the Jth color for points that are J mod K?
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