In mathematics, a Beatty sequence (or homogeneous Beatty sequence) is the sequence of integers found by taking the floor of the positive multiples of an irrational number that is greater than one. 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

that they generate form a pair of complementary Beatty sequences. Here, "complementary" means that every positive integer belongs to exactly one of these two sequences.

Examples

is the golden ratio have the approximate values , known as the lower Wythoff sequence, which is Its integer multiples have the approximate values produces the upper Wythoff sequence, Every positive integer is in exactly one of these two sequences. These sequences define the optimal strategy for Wythoff's game, and are used in the definition of the Wythoff array.

As another example, for the square root of 2. In this case, the sequences are , 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. However, even earlier, in 1894 such sequences were briefly mentioned by Lord Rayleigh in the second edition of his book The Theory of Sound.

Rayleigh theorem

Rayleigh's theorem (also known as Beatty's theorem partition the set of positive integers: each positive integer belongs to exactly one of the two sequences.

First proof

when they are jointly listed in nondecreasing order for positive integers j and k.

for some j and k, a rational number not a rational number. Therefore, no two of the numbers occupy the same position.

.

, 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

: Suppose that, contrary to the theorem, there are integers j>0 and k and m

For non-zero j, the irrationality of r and s

which is impossible (one cannot have an integer between two adjacent integers). Thus the supposition must be false.

: Suppose that, contrary to the theorem, there are integers j>0 and k and m

Since j+1 is non-zero and r and s

which is also impossible. Thus the supposition is false.

Properties

.

.

Relation with Sturmian sequences

The first difference is a characteristic Sturmian word.

Generalizations

.

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 That is, there is no equivalent of Rayleigh's theorem for three or more Beatty sequences.

References

Further reading

• Includes many references.

External links

• Alexander Bogomolny, Beatty Sequences, Cut-the-knot