Allan writes "d_3 = d_4 <= 1/4, but I can't prove equality." Allan's proof is presumably the same as mine: divide the grid into 2x2 blocks and apply the pigeonhole principle. On the other hand, the set 2Z x 2Z has the required property, which gives the reverse inequality, for both n=3 and n=4. For d(5), we get the upper bound 1/5 by dividing the grid into 5-vertex subgraphs shaped like plus-signs (and applying pigeonhole), and the lower bound 1/5 by taking the sublattice of ZxZ spanned by (1,2) and (2,-1). (Right?) What is the first value of n for which it becomes hard to get a pigeonhole-principle upper bound and a doubly-periodic-set lower bound that match? Jim On Tuesday, March 12, 2013, Allan Wechsler <acwacw@gmail.com> wrote:
Focussing on small n rather than on the asymptotics, I get d_1 = 1, d_2 = 1/2 (and can prove both of those). d_3 = d_4 <= 1/4, but I can't prove equality. For n = 5 I can see a pattern with density 1/5, and I think this generalizes to all n that are the sum of two squares (though I can't prove it can be bettered).
On Tue, Mar 12, 2013 at 5:32 PM, James Propp <jamespropp@gmail.com> wrote:
Let d_n be the greatest density achieved by any subset of Z^2, no two points of which are closer in Euclidean distance than sqrt(n). What is known about d_n? Is it known to be always rational? I can use a compactness argument to show that the supremum density is achieved (that's why I felt free to write "the greatest density" above) but I can't prove that it's achieved by a doubly-periodic set (which would imply rational density).
I get n d_n approx 2/sqrt(3) for large n using a back-of-the-envelope calculation, but I may have mis-programmed the envelope, especially since it's not an envelope but rather a bag for a pizza-slice-to-go --- a medium with no track-record as a calculation-aid.
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