Thue–Morse sequence

See also: Prouhet–Thue–Morse constant

This graphic demonstrates the repeating and complementary makeup of the Thue–Morse sequence.

5 logical matrices that give the beginning of the T.-M. sequence, when read line by line

Either in set A (vertical index) or in set B (horizontal index)
is an odd number of elements.

In mathematics, the Thue–Morse sequence, or Prouhet–Thue–Morse sequence, is the binary sequence (an infinite sequence of 0s and 1s) obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far. This procedure yields 0 then 01, 0110, 01101001, 0110100110010110, and so on. The infinite sequence begins:

01101001100101101001011001101001.... (sequence A010060 in the OEIS)

Any other ordered pair of symbols may be used instead of 0 and 1; the logical structure of the Thue–Morse sequence does not depend on the symbols that are used to represent it.

Definition

There are several equivalent ways of defining the Thue–Morse sequence.

Direct definition

To compute the n^th element t_n, write the number n in binary. If the number of ones in this binary expansion is odd then t_n = 1, if even then t_n = 0. For this reason John H. Conway et al. call numbers n satisfying t_n = 1 odious (for odd) numbers and numbers for which t_n = 0 evil (for even) numbers.

Recurrence relation

The Thue–Morse sequence is the sequence t_n satisfying

t0	= 0,
t2n	= tn, and
t2n+1	= 1 − tn.

for all positive integers n.

L-system

The Thue–Morse sequence is a morphic word: it is the output of the following Lindenmayer system:

  variables  0  1
  constants  none
  start      0
  rules      (0 → 01), (1 → 10)

Characterization using bitwise negation

The Thue–Morse sequence in the form given above, as a sequence of bits, can be defined recursively using the operation of bitwise negation. So, the first element is 0. Then once the first 2ⁿ elements have been specified, forming a string s, then the next 2ⁿ elements must form the bitwise negation of s. Now we have defined the first 2ⁿ⁺¹ elements, and we recurse.

Spelling out the first few steps in detail:

• We start with 0.
• The bitwise negation of 0 is 1.
• Combining these, the first 2 elements are 01.
• The bitwise negation of 01 is 10.
• Combining these, the first 4 elements are 0110.
• The bitwise negation of 0110 is 1001.
• Combining these, the first 8 elements are 01101001.
• And so on.

So

• T₀ = 0.
• T₁ = 01.
• T₂ = 0110.
• T₃ = 01101001.
• T₄ = 0110100110010110.
• T₅ = 01101001100101101001011001101001.
• T₆ = 0110100110010110100101100110100110010110011010010110100110010110.
• And so on.

Infinite product

The sequence can also be defined by:

$\prod _{i=0}^{\infty }(1-x^{2^{i}})=\sum _{j=0}^{\infty }(-1)^{t_{j}}x^{j}{\mbox{,}}\!$

where t_j is the j^th element if we start at j = 0.

Some properties

Nested squares generated by successive iterations of Thue–Morse

Because each new block in the Thue–Morse sequence is defined by forming the bitwise negation of the beginning, and this is repeated at the beginning of the next block, the Thue–Morse sequence is filled with squares: consecutive strings that are repeated. That is, there are many instances of XX, where X is some string. However, there are no cubes: instances of XXX. There are also no overlapping squares: instances of 0X0X0 or 1X1X1. The critical exponent is 2.

Notice that T_2n is palindrome for any n > 1. Further, let q_n be a word obtain from T_2n by counting ones between consecutive zeros. For instance, q₁ = 2 and q₂ = 2102012 and so on. The words T_n do not contain overlapping squares in consequence, the words q_n are palindrome squarefree words.

The statement above that the Thue–Morse sequence is "filled with squares" can be made precise: It is a uniformly recurrent word, meaning that given any finite string X in the sequence, there is some length n_X (often much longer than the length of X) such that X appears in every block of length n. The easiest way to make a recurrent sequence is to form a periodic sequence, one where the sequence repeats entirely after a given number m of steps. Then n_X can be set to any multiple of m that is larger than twice the length of X. But the Morse sequence is uniformly recurrent without being periodic, not even eventually periodic (meaning periodic after some nonperiodic initial segment).

We define the Thue–Morse morphism to be the function f from the set of binary sequences to itself by replacing every 0 in a sequence with 01 and every 1 with 10. Then if T is the Thue–Morse sequence, then f(T) is T again; that is, T is a fixed point of f. The function f is a prolongable morphism on the free monoid {0,1}^∗ with T as fixed point: indeed, T is essentially the only fixed point of f; the only other fixed point is the bitwise negation of T, which is simply the Thue–Morse sequence on (1,0) instead of on (0,1). This property may be generalized to the concept of an automatic sequence.

The generating series of T over the binary field is the formal power series

$t(Z)=\sum _{n=0}^{\infty }T(n)Z^{n}\ .$

This power series is algebraic over the field of formal power series, satisfying the equation

$Z+(1+Z)^{2}t+(1+Z)^{3}t^{2}\ .$

In combinatorial game theory

The set of evil numbers (numbers $n$ with $t_{n}=0$) forms a subspace of the nonnegative integers under nim-addition (bitwise exclusive or). For the game of Kayles, the evil numbers form the sparse space—the subspace of nim-values which occur for few (finitely many) positions in the game—and the odious numbers are the common coset.

The Prouhet–Tarry–Escott problem

The Prouhet–Tarry–Escott problem can be defined as: given a positive integer N and a non-negative integer k, partition the set S = { 0, 1, ..., N-1 } into two disjoint subsets S₀ and S₁ that have equal sums of powers up to k, that is:

$\sum _{x\in S_{0}}x^{i}=\sum _{x\in S_{1}}x^{i}$ for all integers i from 1 to k.

This has a solution if N is a multiple of 2^k+1, given by:

• S₀ consists of the integers n in S for which t_n = 0,
• S₁ consists of the integers n in S for which t_n = 1.

For example, for N = 8 and k = 2,

0 + 3 + 5 + 6 = 1 + 2 + 4 + 7,

0² + 3² + 5² + 6² = 1² + 2² + 4² + 7².

The condition requiring that N be a multiple of 2^k+1 is not strictly necessary: there are some further cases for which a solution exists. However, it guarantees a stronger property: if the condition is satisfied, then the set of k^th powers of any set of N numbers in arithmetic progression can be partitioned into two sets with equal sums. This follows directly from the expansion given by the binomial theorem applied to the binomial representing the n^th element of an arithmetic progression.

Fractals and Turtle graphics

A Turtle Graphics is the curve that is generated if an automaton is programmed with a sequence. If the Thue–Morse Sequence members are used in order to select program states:

• If t(n) = 0, move ahead by one unit,
• If t(n) = 1, rotate counterclockwise by an angle of π/3,

the resulting curve converges to the Koch snowflake, a fractal curve of infinite length containing a finite area. This illustrates the fractal nature of the Thue–Morse Sequence.

Application to fair division

In their book on the problem of fair division, Steven Brams and Alan D. Taylor invoked the Thue–Morse sequence but did not identify it as such. When allocating a contested pile of items between two parties who agree on the items' relative values, Brams and Taylor suggested a method they called balanced alternation, or taking turns taking turns taking turns, as a way to circumvent the favoritism inherent when one party chooses before the other.

Lionel Levine and Katherine Stange, in their discussion of how to fairly share a meal, proposed the Thue–Morse sequence as a way to reduce the advantage of moving first. They suggested that “it would be interesting to quantify the intuition that the Thue-Morse order tends to produce a fair outcome.”

Robert Richman addressed this question, but he too did not identify the Thue–Morse sequence as such at the time of publication. He presented the sequences labeled T_n above as step functions on the interval [0,1] and described their relationship to the Walsh and Rademacher functions. He showed that the n^th derivative can be expressed in terms of T_n. As a consequence, the step function arising from T_n is orthogonal to polynomials of order n − 1. A consequence of this result is that a resource whose value is expressed as a continuous function is most fairly allocated using a sequence that converges to Thue-Morse as the function becomes flatter. An example showed how to pour cups of coffee of equal strength from a carafe with a concentration gradient, prompting a whimsical article in the popular press.

Joshua Cooper and Aaron Dutle showed why the Thue-Morse order provides a fair outcome for discrete events. They considered the fairest way to stage a Galois duel, in which each of the shooters has equally poor shooting skills. Cooper and Dutle postulated that each dueler would demand a chance to fire as soon as the other’s a priori probability of winning exceeded their own. They proved that as the duelers’ hitting probability approaches zero, the firing sequence converges to the Thue-Morse sequence.

Sports competitions form an important class of fair division problems, because sequences of strict alternation often give an unfair advantage to one team. Richman suggested that the fairest way for “captain A” and “captain B” to choose sides for a pick-up game of basketball mirrors T₃: captain A has the first, fourth, sixth, and seventh choices, while captain B has the second, third, fifth, and eighth choices. Ignacio Palacios-Huerta proposed using the Thue–Morse sequence to increase the fairness of various tournament competitions. For example, he suggested that the ex post fairness of a penalty shoot-out in soccer would be improved by changing the sequential order from strict alternation to Thue-Morse.

History

The Thue–Morse sequence was first studied by Eugène Prouhet in 1851, who applied it to number theory. However, Prouhet did not mention the sequence explicitly; this was left to Axel Thue in 1906, who used it to found the study of combinatorics on words. The sequence was only brought to worldwide attention with the work of Marston Morse in 1921, when he applied it to differential geometry. The sequence has been discovered independently many times, not always by professional research mathematicians; for example, Max Euwe, a chess grandmaster, who held the world championship title from 1935 to 1937, and mathematics teacher, discovered it in 1929 in an application to chess: by using its cube-free property (see above), he showed how to circumvent a rule aimed at preventing infinitely protracted games by declaring repetition of moves a draw.

References

• Allouche, Jean-Paul; Shallit, Jeffrey (2003). Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. Zbl 1086.11015.
• Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. (2009). Combinatorics on words. Christoffel words and repetitions in words. CRM Monograph Series. Vol. 27. Providence, RI: American Mathematical Society. ISBN 978-0-8218-4480-9. Zbl 1161.68043.
• Bugeaud, Yann (2012). Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics. Vol. 193. Cambridge: Cambridge University Press. ISBN 978-0-521-11169-0. Zbl pre06066616. {{cite book}}: Check |zbl= value (help)
• Lothaire, M. (2011). Algebraic combinatorics on words. Encyclopedia of Mathematics and Its Applications. Vol. 90. With preface by Jean Berstel and Dominique Perrin (Reprint of the 2002 hardback ed.). Cambridge University Press. ISBN 978-0-521-18071-9. Zbl 1221.68183.
• Lothaire, M. (2005). Applied combinatorics on words. Encyclopedia of Mathematics and Its Applications. Vol. 105. A collective work by Jean Berstel, Dominique Perrin, Maxime Crochemore, Eric Laporte, Mehryar Mohri, Nadia Pisanti, Marie-France Sagot, Gesine Reinert, Sophie Schbath, Michael Waterman, Philippe Jacquet, Wojciech Szpankowski, Dominique Poulalhon, Gilles Schaeffer, Roman Kolpakov, Gregory Koucherov, Jean-Paul Allouche and Valérie Berthé. Cambridge: Cambridge University Press. ISBN 0-521-84802-4. Zbl 1133.68067.
• Pytheas Fogg, N. (2002). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. Vol. 1794. Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. Berlin: Springer-Verlag. ISBN 3-540-44141-7. Zbl 1014.11015.

External links

• Weisstein, Eric W. "Thue-Morse Sequence". MathWorld.
• Allouche, J.-P.; Shallit, J. O. The Ubiquitous Prouhet-Thue-Morse Sequence. (contains many applications and some history)
• Thue–Morse Sequence over (1,2) (sequence A001285 in the OEIS)
• Odious numbers (sequence A000069 in the OEIS)
• Evil numbers (sequence A001969 in the OEIS)
• When Thue-Morse meets Koch. A paper showing an astonishing similarity between the Thue–Morse Sequence and the Koch snowflake
• Reducing the influence of DC offset drift in analog IPs using the Thue-Morse Sequence. A technical application of the Thue–Morse Sequence
• MusiNum - The Music in the Numbers. Freeware to generate self-similar music based on the Thue–Morse Sequence and related number sequences.