Rank-3 scale theorems: Difference between revisions

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==== Definitions: LQ (WIP) ====

First attempt at a definition: "A scale word ''S'' with ''k'' step sizes X_1, ..., X_k (with a_1 X_1's, ..., and a_k X_k's) is ''line-quantizing'' (LQ) if ''S'', when viewed as a set of instructions tracing a path in Z^k from the origin (in which each X_i means "go 1 step in the positive x_i direction"), results in a path that is a closest approximation to the line [a_1 : a_2 : ... : a_k] intersecting the origin in R^k."

First attempt at a definition: "A scale word ''S'' with ''k'' step sizes X_1, ..., X_k (with a_1 X_1's, ..., and a_k X_k's) is ''line-quantizing'' (LQ) if ''S'', when viewed as a set of instructions tracing a path in Z^k from the origin (in which each X_i means "go 1 step in the positive x_i direction"), results in a path γ(S) that is a closest approximation to the line [a_1 : a_2 : ... : a_k] intersecting the origin in R^k."

The problem with this definition is that it is unclear what "closest approximation" means. We take several different definitions of that phrase and define different versions of the LQ property.

Let n = a+b+c be the scale size, w = aX bY cZ be the scale word, let R be the corresponding path following the word w (R(k) = your location after taking k steps according to w), and put n+1 equally spaced points p_n on the line segment L = {(a,b,c)t : 0 <= t <= n}, i.e. the points {L(k) = (a,b,c) k : k ∈ {0, ..., n}}.

Let n = a+b+c be the scale size, w = aX bY cZ be the scale word, let γ(w) be the corresponding path following the word w (γ(w)(k) = your location after taking k steps according to w), and put n+1 equally spaced points p_n on the line segment L = {(a,b,c)t : 0 <= t <= n}, i.e. the points {L(k) = (a,b,c) k : k ∈ {0, ..., n}}.

* Assume ''S'' is a 2-step scale. Then ''S'' is ''slope-LQ'' if the slope between any two pair of points (representing a ''k''-mosstep) is one of the two nearest possible slopes (in the set {k/0,...,0/k}) to b/a.

* Say that a 2-step scale ''S'' is ''floor-LQ'' if some mode of ''S'' ~~gives~~ the graph of floor(b/a*x).

* Say that a 2-step scale ''S'' is ''floor-LQ'' if some mode ''M'' of ''S'' satisfies that γ(''M'') = the graph of floor(b/a*x).

* Say that a k-step scale ''S'' is ''LQ'' if ...

* Say that a k-step scale ''S'' is ''LQ'' if γ(''S'') is a translate of round({(a,b,c)t : 0 <= t <= 1}) (where round = round each coordinate up).

===== MV2 is equivalent to floor-LQ in 2-step scales (WIP) =====

Assume wlog there are more L's than s's.

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 ==== Definitions: LQ (WIP) ====
-First attempt at a definition: "A scale word ''S'' with ''k'' step sizes X_1, ..., X_k (with a_1 X_1's, ..., and a_k X_k's) is ''line-quantizing'' (LQ) if ''S'', when viewed as a set of instructions tracing a path in Z^k from the origin (in which each X_i means "go 1 step in the positive x_i direction"), results in a path that is a closest approximation to the line [a_1 : a_2 : ... : a_k] intersecting the origin in R^k."
+First attempt at a definition: "A scale word ''S'' with ''k'' step sizes X_1, ..., X_k (with a_1 X_1's, ..., and a_k X_k's) is ''line-quantizing'' (LQ) if ''S'', when viewed as a set of instructions tracing a path in Z^k from the origin (in which each X_i means "go 1 step in the positive x_i direction"), results in a path γ(S) that is a closest approximation to the line [a_1 : a_2 : ... : a_k] intersecting the origin in R^k."
 The problem with this definition is that it is unclear what "closest approximation" means. We take several different definitions of that phrase and define different versions of the LQ property.
-Let n = a+b+c be the scale size, w = aX bY cZ be the scale word, let R be the corresponding path following the word w (R(k) = your location after taking k steps according to w), and put n+1 equally spaced points p_n on the line segment L = {(a,b,c)t : 0 <= t <= n}, i.e. the points {L(k) = (a,b,c) k : k ∈ {0, ..., n}}.
+Let n = a+b+c be the scale size, w = aX bY cZ be the scale word, let γ(w) be the corresponding path following the word w (γ(w)(k) = your location after taking k steps according to w), and put n+1 equally spaced points p_n on the line segment L = {(a,b,c)t : 0 <= t <= n}, i.e. the points {L(k) = (a,b,c) k : k ∈ {0, ..., n}}.
 * Assume ''S'' is a 2-step scale. Then ''S'' is ''slope-LQ'' if the slope between any two pair of points (representing a ''k''-mosstep) is one of the two nearest possible slopes (in the set {k/0,...,0/k}) to b/a.
-* Say that a 2-step scale ''S'' is ''floor-LQ'' if some mode of ''S'' gives the graph of floor(b/a*x).
+* Say that a 2-step scale ''S'' is ''floor-LQ'' if some mode ''M'' of ''S'' satisfies that γ(''M'') = the graph of floor(b/a*x).
-* Say that a k-step scale ''S'' is ''LQ'' if ...
+* Say that a k-step scale ''S'' is ''LQ'' if γ(''S'') is a translate of round({(a,b,c)t : 0 <= t <= 1}) (where round = round each coordinate up).
 ===== MV2 is equivalent to floor-LQ in 2-step scales (WIP) =====
 Assume wlog there are more L's than s's.