Looks like you can't do it over an alphabet of 2 letters (use breadth-first search) but you can do it over an alphabet of 3 letters. Take the Thue-Morse word, replace each 0 by 010102 and 1 by 010202 and let {a} be the vowels and {1,2} be consonants and r = 1/2. Another construction: take the Thue-Morse word 01101001... and insert a 2 between each letter to get 2021212021202021 ... and let {2} be vowel and {0,1} be consonants. Jeff On 1/29/17 7:56 PM, Andy Latto wrote:
Is the following a counterexample?
Take the Thue sequence, and replace each 1 by (a, b), and each 0 by (c, e). a and e are vowels, b and c are consonants, and r = 1.
Andy
On Sun, Jan 29, 2017 at 7:30 PM, James Propp <jamespropp@gmail.com> wrote:
The following question has arisen from my research on one-dimensional packing; I'm wondering if someone can help me prove it.
Say we have an infinite word W = (w_1, w_2, w_3, ...) consisting of letters from some finite alphabet each of which occurs infinitely often in W. Suppose further that the finite alphabet consists of two kinds of letters which I'll call "consonants" and "vowels". We are also given a rational number r, and our infinite word satisfies a "proportionality property" (P):
(P) If w_i = w_j (with i < j), then the number of vowels in (w_i, w_{i+1}, ..., w_{j-1}) (counted with multiplicity) divided by the number of consonants in (w_i, w_{i+1}, ..., w_{j-1}) (counted with multiplicity) equals r.
Then I would like to be able to conclude that the consonant-vowel pattern of W is eventually periodic.
Example: W = (a, b, c, e, d, c, a, b, c, a, b, c, e, d, c, a, b, c, a, b, c, a, b, c, e, d, c, ....) where a and e are vowels and b, c, and d are consonants. (If the pattern isn't clear, write the word schematically as XY XXY XXXY XXXXY ... where X = (a, b, c) and Y = (e, d, c).) Let r = 1/2. With i = 1 and j = 7, we see that the number of vowels in positions 1 through 6 divided by the number of consonants in positions 1 through 6 equals 2/4, or 1/2. More generally, given any two positions i and j that both have the same letter in W, if we look at all the letters that occur in positions i through j-1, we see that the number of vowels divided by the number of consonants is 1/2, so property (P) is satisfied. And if we look at the consonant-vowel pattern, it's just vowel-consonant-consonant over and over again, repeating periodically.
My gut tells me that this is at the level of a hard IMO problem, and that it has a pretty solution. But that's no consolation to me, given that I don't see how to solve it.
A solution to this puzzle would advance me towards a seemingly unrelated goal in the theory of sphere-packing.
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