If I had time I'd start investigating this looking at the Pythagorean triples to get some bounds and insight. These will give periodic grids. We can assume the matching is *also* periodic and see what happens (the periodicity of the matching may or may not match the periodicity of the tiling). Both with and without a coinciding point would be useful. There are effective matching algorithms that will probably work up to a pretty large number of points, especially with a reasonably tight bound on the allowed distance. On Fri, May 29, 2015 at 8:49 AM, Warren D Smith <warren.wds@gmail.com> wrote:
From: Allan Wechsler <acwacw@gmail.com> So the current state of the art is that the minimum possible c is between 0.707... and 0.92... ? I wonder if a computer search on a reasonably-sized piece of Z^2 could give any insights.
--I'm back to not understanding Veit Elser's solution yielding c<0.92... The problem I have is: although the amazing TILE is 8-way rotationally symmetric, the TILING is not, it has only 4-way rotational symmetry, same symmetry as the grid Z^2=G1. In short the tiling "does not know about" G2. What would have established 0.92 is, each tile contains exactly one point in G1 and exactly one point in G2. I'm only seeing one of those claims.
The state of my art is, c is logarithmically infinite or less, almost certainly less. If Elser's result is discarded we currently have no proof c is finitely bounded, but do have nonrigorous arguments, which perhaps can be rigorized, indicating it grows slowly at most, at most like (logN)^(3/4) or logstar(N), and perhaps is finite. Indeed in the Lp norm versions of the problem with p finite, I have given an argument c is finitely bounded, and sketched how to rigorize the argument (but have not carried thru the details, so caveat emptor).
The proof of the O(logN) upper bound is rigorous but only works for rotation angles whose tangent is quadratic irrational.
I think this is an excellent problem. Great rich depths are hidden inside it.
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