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, not-necessarily-linearly-independent step sizes X₁, ..., X_r, and the number of X_i steps in the scale is a_i), 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 (a₁, ..., a_r) given by the scale signature. The particle's trajectory will be periodic, and with probability one, any collision only be with one wall at a time. The pattern of which walls the particle collides with then spells out 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);
• n = a₁ + ... + a_r be the length of w;
• L be a line of the form L(t) = (a₁, ..., a_r)t + v₀, where v₀ is a constant vector in R^r. We say that L is in generic position if L intersects the hyperplane x₁ = 0 at a point (0, α₁, α₂, ... α_r-1) where α_i and α_j/α_i for i ≠ j are irrational.

We call w a rank-r billiard scale if any line in generic position of the form (a₁, ..., a_r)t + v₀ 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.

Properties

Proofs to be added

• Mosses are rank-2 billiard scales
• A billiard scale projects to a billiard scale of lower rank when one removes all instances of some subset of its step sizes
• Given any r and any "nondegenerate" scale signature a₁X₁, ..., a_rX_r (i.e. no a_i is 0), there are only finitely many billiard scales with that signature up to rotation
  • Finiteness is obvious; how does the number of billiard scales with a given signature depend on r or with the signature?