I asked a question similar to Steve's "peppering problem" for the circle on MathOverflow a few years ago. My assumption was also that k phi (mod 1), k = 1, 2, ... would be the way to maximize lim inf [N d(N)], where d(N) is the minimum separation between the first N points. This turned out to be wrong, and you can do substantially better (0.721 vs 0.618 for the golden ratio process). https://mathoverflow.net/questions/275158/sequential-addition-of-points-on-a... On the 2-torus or the sphere for that matter I have no idea what to expect. Yoav On Fri, Sep 11, 2020 at 4:08 PM <math-fun-request@mailman.xmission.com> wrote:
---------- Forwarded message ---------- From: Steve Witham <sw@tiac.net> To: math-fun@mailman.xmission.com Cc: Bcc: Date: Fri, 11 Sep 2020 14:23:10 -0400 Subject: Re: [math-fun] Tammes problem for the square and cubical torus
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But you *could* "pack" points on a unit interval: k phi (mod 1) for k = 1 to N.
...Why would you want to do that!? you ask. Because you can keep adding points to the interval incrementally, and the neighbor distances (mod 1) are always within some constant factor of 1/N.
I have searched but not found a similarly simple way to add points incrementally to the square torus that gets remotely good packing. (I call this the peppering problem.)
--Steve