Beatty sequence

In mathematics, a Beatty sequence (or homogeneous 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 $r\,$ generates the Beatty sequence

$\mathcal{B}_r = \lfloor r \rfloor, \lfloor 2r \rfloor, \lfloor 3r \rfloor,\ldots = ( \lfloor nr \rfloor)_{n\geq 1}$

If $r > 1 \,,$ then $s = r/(r-1)\,$ is also a positive irrational number. They naturally satisfy

$\frac1r + \frac1s = 1 \,$

and the sequences

$\mathcal{B}_r = ( \lfloor nr \rfloor)_{n\geq 1}$ and

$\mathcal{B}_s = ( \lfloor ns \rfloor)_{n\geq 1}$

form a pair of complementary Beatty sequences.

A more general non-homogeneous Beatty sequence takes the form

$\mathcal{B}_r = \lfloor r+p \rfloor, \lfloor 2r+p \rfloor, \lfloor 3r+p \rfloor,\ldots = ( \lfloor nr+p \rfloor)_{n\geq 1}$

where $p\,$ is a real number. For $p=1\,$, the complementary non-homogeneous Beatty sequences can be found by making $t = 1/r\,$ so that

$\mathcal{B}_r = ( \lfloor n(r+1) \rfloor)_{n\geq 1}$ and

$\mathcal{B}_t = ( \lfloor n(t+1) \rfloor)_{n\geq 1}$

form a pair of complementary Beatty sequences.

Examples

For r = the golden mean, we have s = r + 1. In this case, the sequence $( \lfloor nr \rfloor)$, known as the lower Wythoff sequence, is

• 1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, ... (sequence A000201 in OEIS).

and the complementary sequence $( \lfloor ns \rfloor)$, the upper Wythoff sequence, is

• 2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 41, 44, 47, ... (sequence A001950 in OEIS).

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 = √2, we have s = 2 + √2. In this case, the sequences are

• 1, 2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 24, ... (sequence A001951 in OEIS) and
• 3, 6, 10, 13, 17, 20, 23, 27, 30, 34, 37, 40, 44, 47, 51, 54, 58, ... (sequence A001952 in OEIS).

And for r = π and s = π/(π - 1) the sequences are

• 3, 6, 9, 12, 15, 18, 21, 25, 28, 31, 34, 37, 40, 43, 47, 50, 53, ... (sequence A022844 in OEIS) and
• 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 24, 26, ... (sequence A054386 in OEIS).

Notice that any number in the first sequence is lacking 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.^[1][2] 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 (3rd Baron Rayleigh) in the second edition of his book The Theory of Sound.^[3]

Rayleigh theorem

The Rayleigh theorem (also known as Beatty's theorem) states that given an irrational number $r > 1 \,,$ there exists $s > 1$ so that the Beatty sequences $\mathcal{B}_r$ and $\mathcal{B}_s$ partition the set of positive integers: each positive integer belongs to exactly one of the two sequences.^[3]

First proof

Given $r > 1 \,,$ let $s = r/(r-1)\,$. We must show that every positive integer lies in one and only one of the two sequences $\mathcal{B}_r$ and $\mathcal{B}_s$. We shall do so by considering the ordinal positions occupied by all the fractions j/r and k/s 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 (as a single number), suppose to the contrary that $j/r = k/s$ for some j and k. Then r/s = j/k, a rational number, but also, $r/s = r(1 - 1/r) = r - 1,$ not a rational number. Therefore, no two of the numbers occupy the same position.

For any j/r, there are j numbers i/r ≤ j/r and $\lfloor js/r \rfloor$ numbers $k/s \le j/r$, so that the position of $j/r$ in the list is $j + \lfloor js/r \rfloor$. The equation $1/r + 1/s = 1$ implies

$j + \lfloor js/r \rfloor = j + \lfloor j(s - 1) \rfloor = \lfloor js \rfloor.$

Likewise, the position of k/s in the list is $\lfloor kr \rfloor$.

Conclusion: every positive integer (that is, every position in the list) is of the form $\lfloor nr \rfloor$ or of the form $\lfloor ns \rfloor$, 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

$j = \left\lfloor {k \cdot r} \right\rfloor = \left\lfloor {m \cdot s} \right\rfloor \,.$

This is equivalent to the inequalities

$j \le k \cdot r < j + 1 \text{ and } j \le m \cdot s < j + 1. \,$

For non-zero j, the irrationality of r and s is incompatible with equality, so

$j < k \cdot r < j + 1 \text{ and } j < m \cdot s < j + 1 \,$

which lead to

${j \over r} < k < {j + 1 \over r} \text{ and } {j \over s} < m < {j + 1 \over s}. \,$

Adding these together and using the hypothesis, we get

$j < k + m < j + 1 \,$

which is impossible (one cannot have an integer between two adjacent integers). 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

$k \cdot r < j \text{ and } j + 1 \le (k + 1) \cdot r \text{ and } m \cdot s < j \text{ and } j + 1 \le (m + 1) \cdot s \,.$

Since j + 1 is non-zero and r and s are irrational, we can exclude equality, so

$k \cdot r < j \text{ and } j + 1 < (k + 1) \cdot r \text{ and } m \cdot s < j \text{ and } j + 1 < (m + 1) \cdot s. \,$

Then we get

$k < {j \over r} \text{ and } {j + 1 \over r} < k + 1 \text{ and } m < {j \over s} \text{ and } {j + 1 \over s} < m + 1 \,$

Adding corresponding inequalities, we get

$k + m < j \text{ and } j + 1 < k + m + 2 \,$

$k + m < j < k + m + 1 \,$

which is also impossible. Thus the supposition is false.

Properties

$m \in \mathcal{B}_r$ if and only if

$1 - \frac{1}{r} < \left[ \frac{m}{r} \right]_1$

where $[x]_1$ denotes the fractional part of $x$ i.e., $[x]_1 = x - \lfloor x \rfloor$.

Proof: $m \in B_r$ $\Leftrightarrow \exists n, m = \lfloor nr \rfloor$ $\Leftrightarrow m < nr < m + 1$ $\Leftrightarrow \frac{m}{r} < n < \frac{m}{r} + \frac{1}{r}$ $\Leftrightarrow n - \frac{1}{r} < \frac{m}{r} < n$ $\Leftrightarrow 1 - \frac{1}{r} < \left[ \frac{m}{r} \right]_1$

Furthermore, $m = \left\lfloor \left( \left\lfloor \frac{m}{r} \right\rfloor + 1 \right) r \right\rfloor$

Proof: $m = \left\lfloor \left( \left\lfloor \frac{m}{r} \right\rfloor + 1 \right) r \right\rfloor$ $\Leftrightarrow m < \left( \left\lfloor \frac{m}{r} \right\rfloor + 1 \right) r < m + 1$ $\Leftrightarrow \frac{m}{r} < \left\lfloor \frac{m}{r} \right\rfloor + 1 < \frac{m + 1}{r}$ $\Leftrightarrow \left\lfloor \frac{m}{r} \right\rfloor + 1 - \frac{1}{r} < \frac{m}{r} < \left\lfloor \frac{m}{r} \right\rfloor + 1$ $\Leftrightarrow 1 - \frac{1}{r} < \frac{m}{r} - \left\lfloor \frac{m}{r} \right\rfloor =\left[ \frac{m}{r} \right]_1$

Relation with Sturmian sequences

The first difference

$\lfloor (n+1)r\rfloor-\lfloor nr\rfloor$

of the Beatty sequence associated to the irrational number $r$ is a characteristic Sturmian word over the alphabet $\{\lfloor r\rfloor,\lfloor r\rfloor+1\}$.

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 theorem states that, if $\alpha_1,\ldots,\alpha_n$ are positive real numbers such that $(\lfloor k\alpha_i\rfloor)_{k,i\ge1}$ contains all positive integers exactly once, then $n\le2.$ That is, there is no equivalent of Rayleigh's theorem to three or more Beatty sequences.^[4][5]

References

1. Beatty, Samuel; Dunkel, O.; Pelletier, A.; Irwin, F.; Riley, J. L.; Fitch, P.; Yost, D. M. (1926). "Problem 3173". American Mathematical Monthly 33 (3): 159. doi:10.2307/2300153. Missing |last2= in Authors list (help)
2. S. Beatty, A. Ostrowski, J. Hyslop, A. C. Aitken (1927). "Solutions to Problem 3173". American Mathematical Monthly 34 (3): 159–160. doi:10.2307/2298716. JSTOR 2298716.
3. John William Strutt, 3rd Baron Rayleigh (1894). The Theory of Sound 1 (Second ed.). Macmillan. p. 123.
4. J. V. Uspensky, On a problem arising out of the theory of a certain game, Amer. Math. Monthly 34 (1927), pp. 516–521.
5. R. L. Graham, On a theorem of Uspensky, Amer. Math. Monthly 70 (1963), pp. 407–409.

• Holshouser, Arthur; Reiter, Harold (2001). "A generalization of Beatty's Theorem". Southwest Journal of Pure and Applied Mathematics 2: 24–29.
• Stolarsky, Kenneth (1976). "Beatty sequences, continued fractions, and certain shift operators". Canadian Mathematical Bulletin 19 (4): 473–482. doi:10.4153/CMB-1976-071-6. MR 0444558. Includes many references.

External links

• Weisstein, Eric W., "Beatty Sequence", MathWorld.
• Alexander Bogomolny, Beatty Sequences, Cut-the-knot