I looked at bijections like this back in the 1980s, though I never tried to determine the optimum c. As I recall, one nice way to construct such bijections (suggested either by Bill Thurston or Curt McMullen) involved writing the rotation as a product of skew transformations. If two lattices are related by a skew transformation that preserves infinitely many points, then finding a bijection between them that doesn't move points too far reduces to infinitely 1-dimensional matching problems (and matching two 1-dimensional lattices lying on the same line is trivial). I hope that's clear though I fear it's not. Jim Propp On Wednesday, May 27, 2015, David Wilson <davidwwilson@comcast.net> wrote:
Let G1 = Z^2 be the grid of integer points in the plane R^2.
Let G2 be the grid of points in R^2 gotten by rotating G1 by 45° about the origin.
What is the smallest c for which there exists bijection f: G1óG2 with |P-f(P)| ≤ c for all P?
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