Hypercubic billiard word

Billiard scales are motivated by considering a point particle (a "billiard ball") bouncing off walls in a closed cubic room. Given a scale signature a₁X₁ ... a_rX_r (i.e. stipulating that our scale has r distinct step sizes X₁, ..., X_r, and the number of X_i steps in the scale is a_i > 0), we imagine our billiard ball in an r-dimensional cubic room (with side length 1). We first fire off the billiard ball in the direction a = (a₁, ..., a_r) given by the scale signature. For integer a_i, the particle's trajectory will be periodic, and with probability one, the particle will only collide with one wall at a time. The pattern of which walls the particle collides with then spells out a billiard scale of the given signature, though for arity higher than 2, this can yield rotationally inequivalent scales depending on the starting point.

Identifying opposite sides of the cubic room, yields an equivalent and at times more compelling visualization: now the particle always travels with velocity a, and every time a boundary is crossed and the corresponding scale step recorded, the particle reappears on the other side instead of bouncing. Considering the set of lines that do not yield a billiard scale — namely those that have a point that has multiple integer coordinates — yields a partition of the r-torus into finitely many regions each of which gives rise to a billiard scale.

Formal definition

Formally, let w be a scale word with signature a₁X₁ ... a_rX_r (i.e. w is a scale word with a_i-many X_i steps) and let a = (a₁, ..., a_r), which we call the velocity. We call w a rank-r billiard scale if there exists a vector b ∈ R^r such that the line at + b has intersections with coordinate level planes x_i = k ∈ Z that spell out the scale as you move in the positive t direction along that line. (This definition is equivalent to the definition given in terms of a billiard ball in a cubic room.)

Properties

Proofs to be added

• MOS scales are rank-2 billiard scales
• Not all billiard scales are Fokker blocks; blackdye can be checked to be a billiard scale by using the initial position (1, 1/√5, 1/√3), but it is not a Fokker block.
• A billiard scale becomes a billiard scale of lower rank when one removes all instances of some subset of its step sizes. However, the converse is false.
  • That’s because projecting we just remove some of the αs from the list, leaving all remaining ones intact.
• There are only finitely many billiard scales with a given signature up to rotation
  • Finiteness is obvious; how does the number of billiard scales with a given signature depend on r or on the signature?