In mathematics, a Beatty sequence is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are named after Samuel Beatty, who wrote about them in 1926. Rayleigh's theorem, named after Lord Rayleigh, states that the complement of a Beatty sequence, consisting of the positive integers that are not in the sequence, is itself a Beatty sequence generated by a different irrational number. Beatty sequences can also be used to generate Sturmian words.
Definition
A positive irrational number generates the Beatty sequence If then is also a positive irrational number. These two numbers naturally satisfy the equation. The two Beatty sequences they generate, form a pair of complementary Beatty sequences. Here, "complementary" means that every positive integerbelongs toexactly one of these two sequences.
Examples
When r is the golden mean, we have s = r + 1. In this case, the sequence, known as the lower Wythoff sequence, is
These sequences define the optimal strategy for Wythoff's game, and are used in the definition of the Wythoff array As another example, for r = square root of 2|, we have s = 2 + . In this case, the sequences are
Any number in the first sequence is absent in the second, and vice versa.
History
Beatty sequences got their name from the problem posed in the American Mathematical Monthly by Samuel Beatty in 1926. It is probably one of the most often cited problems ever posed in the Monthly. However, even earlier, in 1894 such sequences were briefly mentioned by John W. Strutt in the second edition of his book The Theory of Sound.
Rayleigh theorem
The Rayleigh theorem states that given an irrational number there exists so that the Beatty sequences and partition the set of positive integers: each positive integer belongs to exactly one of the two sequences.
First proof
Given let. We must show that every positive integer lies in one and only one of the two sequences and. We shall do so by considering the ordinal positions occupied by all the fractions and when they are jointly listed in nondecreasing order for positive integers j and k. To see that no two of the numbers can occupy the same position, suppose to the contrary that for some j and k. Then =, a rational number, but also, not a rational number. Therefore, no two of the numbers occupy the same position. For any, there are j numbers and numbers, so that the position of in the list is. The equation implies Likewise, the position of in the list is. Conclusion: every positive integer is of the form or of the form, but not both. The converse statement is also true: if p and q are two real numbers such that every positive integer occurs precisely once in the above list, then p and q are irrational and the sum of their reciprocals is 1.
Second proof
Collisions: Suppose that, contrary to the theorem, there are integers j > 0 and k and m such that This is equivalent to the inequalities For non-zero j, the irrationality of r and s is incompatible with equality, so which lead to Adding these together and using the hypothesis, we get which is impossible. Thus the supposition must be false. Anti-collisions: Suppose that, contrary to the theorem, there are integers j > 0 and k and m such that Since j + 1 is non-zero and r and s are irrational, we can exclude equality, so Then we get Adding corresponding inequalities, we get which is also impossible. Thus the supposition is false.
Properties
if and only if where denotes the fractional part of i.e.,. Proof: Furthermore,. Proof:
Relation with Sturmian sequences
The first difference of the Beatty sequence associated with the irrational number is a characteristic Sturmian word over the alphabet.
Generalizations
If slightly modified, the Rayleigh's theorem can be generalized to positive real numbers and negative integers as well: if positive real numbers and satisfy, the sequences and form a partition of integers. The Lambek–Moser theorem generalizes the Rayleigh theorem and shows that more general pairs of sequences defined from an integer function and its inverse have the same property of partitioning the integers. Uspensky's theorem states that, if are positive real numbers such that contains all positive integers exactly once, then That is, there is no equivalent of Rayleigh's theorem to three or more Beatty sequences.
