On Fri, Mar 23, 2012 at 2:17 AM, Dan Asimov <dasimov@earthlink.net> wrote:

A propos the sunflower:

Let a circle have unit circumference: C  =  R/Z, and let tau be (sqrt(5)-1)/2.

For any sequence S = {x_n in C  |  n=1,2,3,...}, define M_n(S) as the length of the largest arc of C in the complement of {x_1,...,x_n}.

Claim (?): The sequence in C

    S_tau := {x_n := n*tau (mod Z)  |  n=1,2,3,...}

is the (unique ?) sequence S whose maximum gap M_n(S), as a function of n, decreases *in some sense* faster than that of any other sequence in C.

Of course, there are sequences T for which M_n(T) = 1/n (the minimum possible and for n > 1 always < M_n(S_tau)) for an unbounded set of values of n.

SO: Can someone state the claim rigorously and correctly?

Here's a way to define what is meant by the "best" sequence. I have no idea whether S_tau is the best sequence under this definitions, but it might be. Define the best sequence as the one that minimizes lim sup(n M_n(S)) The series that divides the circle into k equal parts, then divides those parts into k equal parts, and so forth, is better than the golden sequence at powers of k, but it's much worse at numbers slightly less than a power of k. The golden ratio sequence is never perfect, but it's never really bad; any sequence which has an infinite series of "bad" values of n will have a larger lim sup. I have no idea how to go about proving that the golden ratio sequence is the best by this measure. But it seems plausible to me. Andy