[1] For example, b4 = 1 because the binary representation of 4 is 100, which only contains one block of consecutive 0s of length 2; whereas b5 = 0 because the binary representation of 5 is 101, which contains a block of consecutive 0s of length 1.
[2] In 1949, Khinchin conjectured that there does not exist a non-quadratic algebraic real number having bounded partial quotients in its continued fraction expansion.
[3][4] Baum and Sweet's paper showed that the same expectation is not met for algebraic power series.
(The degree of the power series in Baum and Sweet's result is analogous to the degree of the field extension associated with the algebraic real in Khinchin's conjecture.)
One of the series considered in Baum and Sweet's paper is a root of The authors show that by Hensel's lemma, there is a unique such root in
, which factors as They go on to prove that this unique root has partial quotients of degree
Before doing so, they state (in the remark following Theorem 2, p 598)[2] that the root can be written in the form where
Mkaouar[6] and Yao[7] proved that the partial quotients of the continued fraction for
[8] However, the sequence of partial quotients can be generated by a non-uniform morphism.
[9] The value of term bn in the Baum–Sweet sequence can be found recursively as follows.
If n = m·4k, where m is not divisible by 4 (or is 0), then Thus b76 = b9 = b4 = b0 = 1, which can be verified by observing that the binary representation of 76, which is 1001100, contains no consecutive blocks of 0s with odd length.
The Baum–Sweet word 1101100101001001..., which is created by concatenating the terms of the Baum–Sweet sequence, is a fixed point of the morphism or string substitution rules as follows: From the morphism rules it can be seen that the Baum–Sweet word contains blocks of consecutive 0s of any length (bn = 0 for all 2k integers in the range 5.2k ≤ n < 6.2k), but it contains no block of three consecutive 1s.
More succinctly, by Cobham's little theorem the Baum–Sweet word can be expressed as a coding
applied to the fixed point of a uniform morphism