Billiard scale

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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 ... arXr (i.e. stipulating that our scale has r distinct, not-necessarily-linearly-independent step sizes X1, ..., Xr, and the number of Xi steps in the scale is ai > 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 (a1, ..., ar) given by the scale signature. For integer ai, 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.

Formal definition

Formally, let

  • w be a scale word with signature a1X1 ... arXr (i.e. w is a scale word with ai-many Xi steps);
  • L be a line of the form L(t) = (a1, ..., ar)t + v0, where v0 is a constant vector in Rr. We say that L is in generic position if L intersects the hyperplane x1 = 0 at a point (0, α1, α2, ... αr-1) where αi and αji for ij are irrational.

We call w a rank-r billiard scale if any line in generic position of the form (a1, ..., ar)t + v0 has intersections with coordinate level planes xi = kZ 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.)


Proofs to be added

  • Mosses are rank-2 billiard scales
  • Fokker blocks are billiard scales. (?)
  • A billiard scale becomes a billiard scale of lower rank when one removes all instances of some subset of its step sizes
    • 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?