Liouville constants #

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This file contains a construction of a family of Liouville numbers, indexed by a natural number $m$. The most important property is that they are examples of transcendental real numbers. This fact is recorded in transcendental_liouville_number.

More precisely, for a real number $m$, Liouville's constant is $$ \sum_{i=0}^\infty\frac{1}{m^{i!}}. $$ The series converges only for $1 < m$. However, there is no restriction on $m$, since, if the series does not converge, then the sum of the series is defined to be zero.

We prove that, for $m \in \mathbb{N}$ satisfying $2 \le m$, Liouville's constant associated to $m$ is a transcendental number. Classically, the Liouville number for $m = 2$ is the one called ``Liouville's constant''.

Implementation notes #

The indexing $m$ is eventually a natural number satisfying $2 ≤ m$. However, we prove the first few lemmas for $m \in \mathbb{R}$.

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noncomputable def liouville_number (m : ℝ) :

ℝ

For a real number m, Liouville's constant is $$ \sum_{i=0}^\infty\frac{1}{m^{i!}}. $$ The series converges only for 1 < m. However, there is no restriction on m, since, if the series does not converge, then the sum of the series is defined to be zero.

Equations

liouville_number m = ∑' (i : ℕ), 1 / m ^ i.factorial

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noncomputable def liouville_number.partial_sum (m : ℝ) (k : ℕ) :

ℝ

liouville_number.partial_sum is the sum of the first k + 1 terms of Liouville's constant, i.e. $$ \sum_{i=0}^k\frac{1}{m^{i!}}. $$

Equations

liouville_number.partial_sum m k = (finset.range (k + 1)).sum (λ (i : ℕ), 1 / m ^ i.factorial)

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noncomputable def liouville_number.remainder (m : ℝ) (k : ℕ) :

ℝ

liouville_number.remainder is the sum of the series of the terms in liouville_number m starting from k+1, i.e $$ \sum_{i=k+1}^\infty\frac{1}{m^{i!}}. $$

Equations

liouville_number.remainder m k = ∑' (i : ℕ), 1 / m ^ (i + (k + 1)).factorial

We start with simple observations.

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@[protected]

theorem liouville_number.summable {m : ℝ} (hm : 1 < m) :

summable (λ (i : ℕ), 1 / m ^ i.factorial)

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theorem liouville_number.remainder_summable {m : ℝ} (hm : 1 < m) (k : ℕ) :

summable (λ (i : ℕ), 1 / m ^ (i + (k + 1)).factorial)

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theorem liouville_number.remainder_pos {m : ℝ} (hm : 1 < m) (k : ℕ) :

0 < liouville_number.remainder m k

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theorem liouville_number.partial_sum_succ (m : ℝ) (n : ℕ) :

liouville_number.partial_sum m (n + 1) = liouville_number.partial_sum m n + 1 / m ^ (n + 1).factorial

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theorem liouville_number.partial_sum_add_remainder {m : ℝ} (hm : 1 < m) (k : ℕ) :

liouville_number.partial_sum m k + liouville_number.remainder m k = liouville_number m

Split the sum definining a Liouville number into the first k term and the rest.

We now prove two useful inequalities, before collecting everything together.

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theorem liouville_number.remainder_lt' (n : ℕ) {m : ℝ} (m1 : 1 < m) :

liouville_number.remainder m n < (1 - 1 / m)⁻¹ * (1 / m ^ (n + 1).factorial)

An upper estimate on the remainder. This estimate works with m ∈ ℝ satisfying 1 < m and is stronger than the estimate liouville_number.remainder_lt below. However, the latter estimate is more useful for the proof.

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theorem liouville_number.aux_calc (n : ℕ) {m : ℝ} (hm : 2 ≤ m) :

(1 - 1 / m)⁻¹ * (1 / m ^ (n + 1).factorial) ≤ 1 / (m ^ n.factorial) ^ n

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theorem liouville_number.remainder_lt (n : ℕ) {m : ℝ} (m2 : 2 ≤ m) :

liouville_number.remainder m n < 1 / (m ^ n.factorial) ^ n

An upper estimate on the remainder. This estimate works with m ∈ ℝ satisfying 2 ≤ m and is weaker than the estimate liouville_number.remainder_lt' above. However, this estimate is more useful for the proof.

Starting from here, we specialize to the case in which m is a natural number.

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theorem liouville_number.partial_sum_eq_rat {m : ℕ} (hm : 0 < m) (k : ℕ) :

∃ (p : ℕ), liouville_number.partial_sum ↑m k = ↑p / ↑m ^ k.factorial

The sum of the k initial terms of the Liouville number to base m is a ratio of natural numbers where the denominator is m ^ k!.

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theorem liouville_liouville_number {m : ℕ} (hm : 2 ≤ m) :

liouville (liouville_number ↑m)

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theorem transcendental_liouville_number {m : ℕ} (hm : 2 ≤ m) :

transcendental ℤ (liouville_number ↑m)