Proof sketch for polyomino packing density upper bound

Brief elaboration on the upper bound argument:

Let P be the heptomino given in the post, and R the 65-celled region pictured.

Given a packing of P, let delta_n be the fraction of cells covered by P on the region [-n,n] x [-n,n]. We define the density of a packing of P to be the limit superior of delta_n as n goes to infinity.

Suppose there is a packing of P with a density d = 64/65+epsilon for some epsilon>0. Then there exist arbitrarily large N such that delta_N > 64/65 + epsilon/2.

Now, choose a random point A from among the 4(N+11)^2 in [-N-11, N+11] x [-N-11, N+11], set this point as the center of our region, and choose a second random point B from within the translated region R. The resulting distribution on B is constant on [-N,N] x [-N,N] (because we selected A far enough from the NxN square to avoid boundary conditions with R), and B will be inside the central 2Nx2N square with probability at least ((N-11)/(N+11))^2 (i.e. if A is 11 units inside the border of the NxN square).

So P(the cell at B is covered by P in the packing) >= ((N-11)/(N+11))^2 * delta_N > ((N-11)/(N+11))^2 * (64/65 + epsilon/2)

But we also know that every choice of A must have at least one hole in the resulting translate of R, so P(the cell at B is covered by P in the packing) <= 64/65.

But if we choose N large enough, this obviously leads to a contradiction. Hence, no such packing of density greater than 64/65 can exist.