Lambek–Moser theorem

In combinatorial number theory, the Lambek–Moser theorem defines a partition of the positive integers into two subsets from any non-decreasing integer-valued function and its inverse function. It is a generalization of Rayleigh's theorem, on the complementary Beatty sequences obtained by rounding down irrational multiples of the integers. If a formula is known for the $n$th smallest member of a set of positive integers, the Lambek–Moser theorem can be used to derive a formula for the $n$th positive integer not in the set.

Any partition of the positive integers into two complementary subsets may be defined from a non-decreasing function using the Lambek–Moser theorem. As well as the Beatty sequences, other pairs of complementary sets of positive integers to which the Lambek–Moser theorem can be applied include the even numbers and the odd numbers, the evil numbers and the odious numbers, the prime numbers and the non-primes (1 and the composite numbers), and the $k$th powers and non-powers.

The Lambek–Moser theorem is named for Joachim Lambek and Leo Moser, who published it in 1954, and should be distinguished from an unrelated theorem of Lambek and Moser, later strengthened by Wild, on the number of primitive Pythagorean triples.

Statement

There are two parts to the Lambek–Moser theorem. One part states that two integers functions that are inverse, in a certain sense, can be used to construct a partition of the positive integers into two complementary subsets, and the other part states that every complementary partition can be constructed in this way.

From functions to partitions

The four functions $f$, $f^{*}$, $F$, and $F^{*}$, for the two Beatty sequences 1, 3, 4, 6, ... and 2, 5, 7, 10, ... . These sequences round down the integer multiples of $\varphi$ and $\varphi +1$, where $\varphi$ is the golden ratio.

The theorem applies to any non-decreasing and unbounded function $f$ that maps positive integers to non-negative integers. From any such function $f$, define $f^{*}$ to be the integer-valued function that is as close as possible to the inverse function of $f$, in the sense that, for all positive integers $n$, $$f{\bigl (}f^{*}(n){\bigr )}<n\leq f{\bigl (}f^{*}(n)+1{\bigr )}.$$ Equivalently, $f^{*}(n)$ may be defined as the number of values $x$ for which $f(x)<n$. It follows from either of these definitions that $f^{*}{}^{*}=f$. Further, define two more functions $F$ and $F^{*}$, from positive integers to positive integers, by $${\begin{aligned}F(n)&=f(n)+n\\F^{*}(n)&=f^{*}(n)+n\\\end{aligned}}$$ Then the first part of the theorem states that $F$ and $F^{*}$ are strictly increasing and that the ranges of $F$ and $F^{*}$ form a partition of the positive integers. Each of these two functions maps any integer $i$ to the $i$th element of its set in the partition.

From partitions to functions

The second part of the Lambek–Moser theorem states that this construction of partitions from inverse functions is universal, in the sense that it can explain any partition of the positive integers into two infinite parts. If $S=s_{1},s_{2},\dots$ and $S^{*}=s_{1}^{*},s_{2}^{*},\dots$ are any two complementary increasing sequences of integers, one may construct a pair of functions $f$ and $f^{*}$ from which this partition may be derived using the Lambek–Moser theorem. To do so, define $f(n)=s_{n}-n$ and $f^{*}(n)=s_{n}^{*}-n$.

Limits of iterated inverses

In the same work in which they proved the Lambek–Moser theorem, Lambek and Moser provided a method of directly constructing $F^{*}$, the mapping from numbers $n$ to the $n$th non-member of a set whose $n$th member is given by $F(n)$, without going through $f$ and $f^{*}$. Let $F^{\#}(n)$ denote the number of values of $x$ for which $F(x)\leq n$; this is an approximation to the inverse function of $F$, but (because it uses $\leq$ in place of $<$) offset by one from the type of inverse used to define $f^{*}$ from $f$. Then, for any $n$, $F^{*}(n)$ is the limit of the sequence $$n,n+F^{\#}(n),n+F^{\#}{\bigl (}n+F^{\#}(n){\bigr )},\dots ,$$ meaning that this sequence eventually becomes constant and the value it takes when it does is $F^{*}(n)$.

For non-negative integers

A variation of the theorem applies to partitions of the non-negative integers, rather than to partitions of the positive integers. For this variation, every partition corresponds to a Galois connection of the ordered non-negative integers to themselves. This is a pair of non-decreasing functions $(f,f^{*})$ with the property that, for all $x$ and $y$, $f(x)\leq y$ if and only if $x\leq f(y)$. The corresponding functions $F$ and $F^{*}$ are defined slightly less symmetrically by $F(n)=f(n)+n$ and $F^{*}(n)=f^{*}(n)+n+1$. For functions defined in this way, the values of $F$ and $F^{*}$ (for non-negative arguments, rather than positive arguments) form a partition of the non-negative integers, and every partition can be constructed in this way.

Examples

For instance, consider the partition of positive integers into even and odd numbers, let $F(n)=2n$ be the $n$th even positive integer, and let $F^{*}(n)=2n-1$ be the $n$th odd positive integer. Then the two functions $f(n)=F(n)-n=n$ and $f^{*}(n)=F^{*}(n)-n=n-1$ form an inverse pair, and the partition generated via the Lambek–Moser theorem from this pair is just the partition of the positive integers into even and odd numbers. Another integer partition, into evil numbers and odious numbers (by the parity of the binary representation) uses almost the same functions, adjusted by the values of the Thue–Morse sequence.

As another example, let $f(n)=n^{2}$, the function that squares its argument. Then its inverse is the square root function, whose closest integer approximation (in the sense used for the Lambek–Moser theorem) is $f^{*}(n)=\lfloor {\sqrt {n-1}}\rfloor$. These two functions give $F(n)=n^{2}+n$ and $F^{*}(n)=\lfloor {\sqrt {n-1}}\rfloor +n.$ For $n=1,2,3,\dots$ the values of $F$ are the pronic numbers

2, 6, 12, 20, 30, 42, 56, 72, 90, 110, ...

while the values of $F^{*}$ are

1, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, ....

These two sequences are complementary: each positive integer belongs to exactly one of them. The Lambek–Moser theorem states that this phenomenon is not specific to the pronic numbers, but rather it arises for any choice of $f$ with the appropriate properties.

The two functions $f$ (rightward blue arrows) and $f^{*}$ (leftward red arrows) arising from the partition of positive integers into primes (2, 3, 5, 7, ...) and non-primes (1, 4, 6, 8, ...). Visualization based on a method of Angel.

Following earlier work of Viggo Brun and D. H. Lehmer, Lambek and Moser discuss formulas involving the prime-counting function for the functions $f$ and $f^{*}$ arising in this way from the partition of the positive integers into prime numbers and non-primes (1 and the composite numbers). If $\pi (n)$ is the prime-counting function (number of primes less than or equal to $n$), then the $n$th non-prime is given by the limit of the sequence $$n,n+\pi (n),n+\pi {\bigl (}n+\pi (n){\bigr )},\dots$$

For some other sequences of integers, the corresponding limit converges in a fixed number of steps, and a direct formula for the complementary sequence is possible. In particular, the $n$th positive integer that is not a $k$th power can be obtained from the limiting formula as $$n+\left\lfloor {\sqrt[{k}]{n+\lfloor {\sqrt[{k}]{n}}\rfloor }}\right\rfloor .$$

Rayleigh's theorem

Main article: Beatty sequence

Rayleigh's theorem states that for two positive irrational numbers $r$ and $s$, both greater than one, with ${\tfrac {1}{r}}+{\tfrac {1}{s}}=1$, the two sequences $\lfloor i\cdot r\rfloor$ and $\lfloor i\cdot s\rfloor$ for $i=1,2,3,\dots$, obtained by rounding down to an integer the multiples of $r$ and $s$, are complementary. It can be seen as an instance of the Lambek–Moser theorem with $f(n)=\lfloor rn\rfloor -n$ and $f^{\ast }(n)=\lfloor sn\rfloor -n$. The condition that $r$ and $s$ be greater than one implies that these two functions are non-decreasing; the derived functions are $F(n)=\lfloor rn\rfloor$ and $F^{*}(n)=\lfloor sn\rfloor .$ The sequences of values of $F$ and $F^{*}$ forming the derived partition are known as Beatty sequences, after Samuel Beatty's 1926 rediscovery of Rayleigh's theorem.

History and proofs

The theorem was discovered by Leo Moser and Joachim Lambek, who published it in 1954. Moser and Lambek cite the previous work of Samuel Beatty on Beatty sequences as their inspiration, and also cite the work of Viggo Brun and D. H. Lehmer from the early 1930s on methods related to their limiting formula for $F^{*}$. Edsger W. Dijkstra has provided a visual proof of the result, and later another proof based on algorithmic reasoning. Yuval Ginosar has provided an intuitive proof based on an analogy of two athletes running in opposite directions around a circular racetrack.

See also

• Hofstadter Figure-Figure sequences, another pair of complementary sequences to which the Lambek–Moser theorem can be applied

References

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