Approximation exponent

In number theory, a Liouville number is an irrational number x with the property that, for every positive integer n, there exist integers p and q with q > 1 and such that

A Liouville number can thus be approximated "quite closely" by a sequence of rational numbers. In 1844, Joseph Liouville showed that all Liouville numbers are transcendental, thus establishing the existence of transcendental numbers for the first time.

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 (a₁, a₂, …, ), such that a_k ∈ {0, 1, 2, …, b - 1} ∀k ∈ {1, 2, 3, …} and there are infinitely many k with a_k ≠ 0, define the number

In the special case when b = 10, and a_k = 1, ∀k, the resulting number x is called Liouville's constant. The binary Liouville's constant, obtained with b = 2, and a_k = 1, ∀k, is the number

It follows from the definition of x that its base-b representation is

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 q_n and p_n as follows:

Then,

...where the last equality follows from the fact that

Therefore, we conclude that any such x is a Liouville number.

An equivalent definition to the one given above is that for any positive integer n, there exists an infinite number of pairs of integers (p, q ) obeying the above inequality.

Now 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 2^n - 1 > d > 0 (that is, for any integer n > 1 + log₂(d ) ) no pair of integers (p, q ) exists that simultaneously satisfies the two inequalities

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