For small n, over the integers, there is arXiv.org/abs/math.CO/0511636 , but you have to divide the count by 2 as classes by absolute determinant values are enumerated.

Let F_n be the set of positive integers having precisely n prime facfors, counted with multiplicity. For n=1, F_n has a polytime algorithm to decide if a number is in that set or not. For n larger than 1, no such fast algorithm is currently known.

Try XX X, and related, for a series that does not tile the plane without rotation.

Do you mean I has about X/2 integers?

To consider the difficulties, imagine a version where all the equivalence classes have 2 or 3 (or even 1 or 2) members, and consider how little information such a sample will yield.

Not well if many of the equivalence classes are singletons. If you have a nontrivial guarantee on the size of the smallest possible equivalence class, then you may get more confident estimates of the number of classes from a small sample using straightforward calculations.

The verse sounds Carrolian. I may do a poetry search.

It was asked to get an answer. Asking on MO does not mean the answer comes from MO. Sometimes it means the answer comes from the asker.

Granted, many of the examples come from outside mathematics, but pieces like Theoretical Zipperdynamics, by Harry J Zipkin from the Weizpmann Inziptute, should fit your needs.

The Cohen-Kung theorem mentioned in the Wikipedia article suggests to me the possibility that one orientation may cover the torus while the other may not. It might be insightful for the two-by and four-by tori to consider all possible colorings as well as orientations to see which lead to covers and which do not, at least for small values of n.

The website only shows covered squares, not the path that was taken. I was hoping for a series of pictures, one showing a path of 8 steps (with some direction arrows, one of 16 steps, one of 32 steps, and one of 64, showing the ant's trajectory as a red curve superimposed on the grid.

You should assume A is nonsingular. Also, inspired by the idea of looking at a related ratio of the area of a face of a parallelipiped to its volume, I doubt that much more can be said.

I made an erroneous remark, which was partly true. Using a certain indexing of entry and exit points of a ring, one sees that the dynamic induces a parity change between the (indices of) entry and exit points. As a result, the dynamic when translated in ring form will "collapse" the number of possible states to at most 32, and with an even number of rings, half the rings will always be exited at an odd-indexed point, and the other half at an even indexed point. Perhaps this collapse continues with pairs of rings?

Since this torus has a different shape, initial orientation matters. Did you check for both orientations? and are the numbers the same?

This strikes me as too small. In particular, one has a factor of 2^k to account for signs when m is the sum of k nonzero squares. Also, one can pad by zeroes to generate O(k^d) different sets of vectors where each vector has one nonzero component. I expect growth more like exp(m).

Can you supply a picture of the ants path when placed in a 50 by 50 square before the path self-intersects because of the toroidal condition? If r_n is the sequence where r_n steps are taken "before a wraparound effect occurs", perhaps that will suggest asymptotics of s_n.