In mathematics, the Rudin–Shapiro sequence, also known as the Golay–Rudin–Shapiro sequence, is an infinite 2-automatic sequence named after Marcel Golay, Harold S. Shapiro, and Walter Rudin who investigated its properties.

Definition

Each term of the Rudin–Shapiro sequence is either 1 or -1. If the binary expansion of n is given by then let (So u_n is the number of times the block 11 appears in the binary expansion of n.) The Rudin–Shapiro sequence is then defined by Thus r_n = 1 if u_n is even and r_n = -1 if u_n is odd. The sequence u_n is known as the complete Rudin–Shapiro sequence, and starting at n = 0, its first few terms are: and the corresponding terms r_n of the Rudin–Shapiro sequence are: For example, u_6 = 1 and r_6 = -1 because the binary representation of 6 is 110, which contains one occurrence of 11; whereas u_7 = 2 and r_7 = 1 because the binary representation of 7 is 111, which contains two (overlapping) occurrences of 11.

Historical motivation

The Rudin–Shapiro sequence was introduced independently by Golay, Rudin, and Shapiro. The following is a description of Rudin's motivation. In Fourier analysis, one is often concerned with the L^2 norm of a measurable function. This norm is defined by One can prove that for any sequence with each a_n in {1,-1}, Moreover, for almost every sequence with each a_n is in {-1,1}, However, the Rudin–Shapiro sequence satisfies a tighter bound: there exists a constant C > 0 such that It is conjectured that one can take, but while it is known that , the best published upper bound is currently. Let P_n be the n-th Shapiro polynomial. Then, when N = 2^n-1, the above inequality gives a bound on. More recently, bounds have also been given for the magnitude of the coefficients of |P_n(z)|^2 where |z| = 1. Shapiro arrived at the sequence because the polynomials where is the Rudin–Shapiro sequence, have absolute value bounded on the complex unit circle by. This is discussed in more detail in the article on Shapiro polynomials. Golay's motivation was similar, although he was concerned with applications to spectroscopy and published in an optics journal.

Properties

The Rudin–Shapiro sequence can be generated by a 4-state automaton accepting binary representations of non-negative integers as input. The sequence is therefore 2-automatic, so by Cobham's little theorem there exists a 2-uniform morphism \varphi with fixed point w and a coding \tau such that r = \tau(w), where r is the Rudin–Shapiro sequence. However, the Rudin–Shapiro sequence cannot be expressed as the fixed point of some uniform morphism alone. There is a recursive definition The values of the terms rn and un in the Rudin–Shapiro sequence can be found recursively as follows. If n = m·2k where m is odd then Thus u108 = u13 + 1 = u3 + 1 = u1 + 2 = u0 + 2 = 2, which can be verified by observing that the binary representation of 108, which is 1101100, contains two sub-strings 11. And so r108 = (−1)2 = +1. A 2-uniform morphism \varphi that requires a coding \tau to generate the Rudin-Shapiro sequence is the following: The Rudin–Shapiro word +1 +1 +1 −1 +1 +1 −1 +1 +1 +1 +1 −1 −1 −1 +1 −1 ..., which is created by concatenating the terms of the Rudin–Shapiro sequence, is a fixed point of the morphism or string substitution rules as follows: It can be seen from the morphism rules that the Rudin–Shapiro string contains at most four consecutive +1s and at most four consecutive −1s. The sequence of partial sums of the Rudin–Shapiro sequence, defined by with values can be shown to satisfy the inequality Let denote the Rudin–Shapiro sequence on {0,1}, in which case s_n is the number, modulo 2, of occurrences (possibly overlapping) of the block 11 in the base-2 expansion of n. Then the generating function satisfies making it algebraic as a formal power series over. The algebraicity of S(X) over follows from the 2-automaticity of by Christol's theorem. The Rudin–Shapiro sequence along squares is normal. The complete Rudin–Shapiro sequence satisfies the following uniform distribution result. If, then there exists such that which implies that is uniformly distributed modulo 1 for all irrationals x.

Relationship with one-dimensional Ising model

Let the binary expansion of n be given by where. Recall that the complete Rudin–Shapiro sequence is defined by Let Then let Finally, let Recall that the partition function of the one-dimensional Ising model can be defined as follows. Fix N \ge 1 representing the number of sites, and fix constants J > 0 and H > 0 representing the coupling constant and external field strength, respectively. Choose a sequence of weights with each. For any sequence of spins with each, define its Hamiltonian by Let T be a constant representing the temperature, which is allowed to be an arbitrary non-zero complex number, and set where k is the Boltzmann constant. The partition function is defined by Then we have where the weight sequence satisfies \eta_i = 1 for all i.