Liouville number


In number theory, a Liouville number is a real number x with the property that, for every positive integer n, there exist infinitely many pairs of integers with q > 1 such that
Liouville numbers are "almost rational", and can thus be approximated "quite closely" by sequences of rational numbers. They are precisely the transcendental numbers that can be more closely approximated by rational numbers than any algebraic irrational number. In 1844, Joseph Liouville showed that all Liouville numbers are transcendental, thus establishing the existence of transcendental numbers for the first time.

The existence of Liouville numbers (Liouville's constant)

Here we show that Liouville numbers exist by exhibiting a construction that produces such numbers.
For any integer b ≥ 2, and any sequence of integers, such that ak ∈ for all k ∈ and there are infinitely many k with ak ≠ 0, define the number
In the special case when b = 10, and ak = 1, for all k, the resulting number x is called Liouville's constant:
It follows from the definition of x that its base-b representation is
where the nth term is in the th decimal place.
Since this base-b representation is non-repeating it follows that x cannot be rational. Therefore, for any rational number p/q, we have |x − p/q | > 0.
Now, for any integer n ≥ 1, define qn and pn as follows:
Then
Therefore, we conclude that any such x is a Liouville number.

Irrationality

Here we will show that the number x = c/d, where c and d are integers and d > 0, cannot satisfy the inequalities that define a Liouville number. Since every rational number can be represented as such c/d, we will have proven that no Liouville number can be rational.
More specifically, we show that for any positive integer n large enough that 2n − 1 > d > 0 no pair of integers exists that simultaneously satisfies the two inequalities
From this the claimed conclusion follows.
Let p and q be any integers with q > 1. Then we have,
If |cq − dp | = 0, we would have
meaning that such pair of integers would violate the first inequality in the definition of a Liouville number, irrespective of any choice of n.
If, on the other hand, |cq − dp | > 0, then, since cq − dp is an integer, we can assert the sharper inequality |cq − dp | ≥ 1. From this it follows that
Now for any integer n > 1 + log2, the last inequality above implies
Therefore, in the case |cq − dp | > 0 such pair of integers would violate the second inequality in the definition of a Liouville number, for some positive integer n.
We conclude that there is no pair of integers, with q >1, that would qualify such an x = c/d as a Liouville number.
Hence a Liouville number, if it exists, cannot be rational.

Uncountability

Consider, for example, the number
3.1415926...
where the digits are zero except in positions n! where the digit equals the nth digit following the decimal point in the decimal expansion of .
As shown in the section on [|the existence of Liouville numbers], this number, as well as any other non-terminating decimal with its non-zero digits similarly situated, satisfies the definition of a Liouville number. Since the set of all sequences of non-null digits has the cardinality of the continuum, the same thing occurs with the set of all Liouville numbers.
Moreover, the Liouville numbers form a dense subset of the set of real numbers.

Liouville numbers and measure

From the point of view of measure theory, the set of all Liouville numbers L is small. More precisely, its Lebesgue measure, λ, is zero. The proof given follows some ideas by John C. Oxtoby.
For positive integers n > 2 and q ≥ 2 set:
we have
Observe that for each positive integer n ≥ 2 and m ≥ 1, we also have
Since
and n > 2 we have
Now
and it follows that for each positive integer m, L ∩ has Lebesgue measure zero. Consequently, so has L.
In contrast, the Lebesgue measure of the set of all real transcendental numbers is infinite.

Structure of the set of Liouville numbers

For each positive integer n, set
The set of all Liouville numbers can thus be written as
Each Un is an open set; as its closure contains all rationals, it is also a dense subset of real line. Since it is the intersection of countably many such open dense sets, L is comeagre, that is to say, it is a dense Gδ set.

Irrationality measure

The Liouville-Roth irrationality measure of a real number x is a measure of how "closely" it can be approximated by rationals. Generalizing the definition of Liouville numbers, instead of allowing any n in the power of q, we find the largest possible value for μ such that is satisfied by an infinite number of integer pairs with q > 0. This maximum value of μ is defined to be the irrationality measure of x. For any value μ less than this upper bound, the infinite set of all rationals p/q satisfying the above inequality yield an approximation of x. Conversely, if μ is greater than the upper bound, then there are at most finitely many with q > 0 that satisfy the inequality; thus, the opposite inequality holds for all larger values of q. In other words, given the irrationality measure μ of a real number x, whenever a rational approximation x ≅ p/q, p,q ∈ N yields n + 1 exact decimal digits, we have
for any ε>0, except for at most a finite number of "lucky" pairs.
For a rational number α the irrationality measure is μ = 1. The Thue–Siegel–Roth theorem states that if α is an algebraic number, real but not rational, then μ = 2.
Almost all numbers have an irrationality measure equal to 2.
Transcendental numbers have irrationality measure 2 or greater. For example, the transcendental number e has μ = 2. The irrationality measures of, log 2, and log 3 are at most 7.103205334137, 3.57455391, and 5.125, respectively.
It has been proven that if the series converges, then 's irrationality measure is at most 2.5.
The Liouville numbers are precisely those numbers having infinite irrationality measure.

Irrationality base

The irrationality base is a weaker measure of irrationality introduced by J. Sondow, and is regarded as an irrationality measure for Liouville numbers. It is defined as follows:
Let be an irrational number. If there exists a real number with the property that for any, there is a positive integer such that
then is called the irrationality base of and is represented as
If no such exists, then is called a super Liouville number.
Example: The series is a super Liouville number, while the series is a Liouville number with irrationality base 2.

Liouville numbers and transcendence

Establishing that a given number is a Liouville number provides a useful tool for proving a given number is transcendental. However, not every transcendental number is a Liouville number. The terms in the continued fraction expansion of every Liouville number are unbounded; using a counting argument, one can then show that there must be uncountably many transcendental numbers which are not Liouville. Using the explicit continued fraction expansion of e, one can show that e is an example of a transcendental number that is not Liouville. Mahler proved in 1953 that pi| is another such example.
The proof proceeds by first establishing a property of irrational algebraic numbers. This property essentially says that irrational algebraic numbers cannot be well approximated by rational numbers, where the condition for "well approximated" becomes more stringent for larger denominators. A Liouville number is irrational but does not have this property, so it can't be algebraic and must be transcendental. The following lemma is usually known as Liouville's theorem , there being several results known as Liouville's theorem.
Below, we will show that no Liouville number can be algebraic.
Lemma: If α is an irrational number which is the root of a polynomial f of degree n > 0 with integer coefficients, then there exists a real number A > 0 such that, for all integers p, q, with q > 0,
Proof of Lemma: Let M be the maximum value of |f ′| over the interval . Let α1, α2,..., αm be the distinct roots of f which differ from α. Select some value A > 0 satisfying
Now assume that there exist some integers p, q contradicting the lemma. Then
Then p/q is in the interval ; and p/q is not in, so p/q is not a root of f; and there is no root of f between α and p/q.
By the mean value theorem, there exists an x0 between p/q and α such that
Since α is a root of f but p/q is not, we see that |f ′| > 0 and we can rearrange:
Now, f is of the form ci xi where each ci is an integer; so we can express |f| as
the last inequality holding because p/q is not a root of f and the ci are integers.
Thus we have that |f| ≥ 1/qn. Since |f ′| ≤ M by the definition of M, and 1/M > A by the definition of A, we have that
which is a contradiction; therefore, no such p, q exist; proving the lemma.
Proof of assertion: As a consequence of this lemma, let x be a Liouville number; as noted in the article text, x is then irrational. If x is algebraic, then by the lemma, there exists some integer n and some positive real A such that for all p, q
Let r be a positive integer such that 1/ ≤ A. If we let m = r + n, and since x is a Liouville number, then there exist integers a, b where b > 1 such that
which contradicts the lemma. Hence, if a Liouville number exists, it cannot be algebraic, and therefore must be transcendental.