Liouville's theorem

This file contains a proof of Liouville's theorem stating that all Liouville numbers are transcendental.

To obtain this result, there is first a proof that Liouville numbers are irrational and two technical lemmas. These lemmas exploit the fact that a polynomial with integer coefficients takes integer values at integers. When evaluating at a rational number, we can clear denominators and obtain precise inequalities that ultimately allow us to prove transcendence of Liouville numbers.

def Liouville (x : ℝ) : Prop

A Liouville number is a real number x such that for every natural number n, there exist a, b ∈ ℤ with 1 < b such that 0 < |x - a/b| < 1/bⁿ. In the implementation, the condition x ≠ a/b replaces the traditional equivalent 0 < |x - a/b|.

Equations

• Liouville x = ∀ (n : ℕ), ∃ (a : ℤ) (b : ℤ), 1 < b ∧ x ≠ ↑a / ↑b ∧ |x - ↑a / ↑b| < 1 / ↑b ^ n

theorem Liouville.irrational {x : ℝ} (h : Liouville x) : Irrational x

theorem Liouville.exists_one_le_pow_mul_dist {Z : Type u_1} {N : Type u_2} {R : Type u_3} [PseudoMetricSpace R] {d : N → ℝ} {j : Z → N → R} {f : R → R} {α : R} {ε M : ℝ} (d0 : ∀ (a : N), 1 ≤ d a) (e0 : 0 < ε) (B : ∀ ⦃y : R⦄, y ∈ Metric.closedBall α ε → dist (f α) (f y) ≤ dist α y * M) (L : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ Metric.closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))) : ∃ (A : ℝ), 0 < A ∧ ∀ (z : Z) (a : N), 1 ≤ d a * (dist α (j z a) * A)

Let Z, N be types, let R be a metric space, let α : R be a point and let j : Z → N → R be a function. We aim to estimate how close we can get to α, while staying in the image of j. The points j z a of R in the image of j come with a "cost" equal to d a. As we get closer to α while staying in the image of j, we are interested in bounding the quantity d a * dist α (j z a) from below by a strictly positive amount 1 / A: the intuition is that approximating well α with the points in the image of j should come at a high cost. The hypotheses on the function f : R → R provide us with sufficient conditions to ensure our goal. The first hypothesis is that f is Lipschitz at α: this yields a bound on the distance. The second hypothesis is specific to the Liouville argument and provides the missing bound involving the cost function d.

This lemma collects the properties used in the proof of exists_pos_real_of_irrational_root. It is stated in more general form than needed: in the intended application, Z = ℤ, N = ℕ, R = ℝ, d a = (a + 1) ^ f.nat_degree, j z a = z / (a + 1), f ∈ ℤ[x], α is an irrational root of f, ε is small, M is a bound on the Lipschitz constant of f near α, n is the degree of the polynomial f.

theorem Liouville.exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f : Polynomial ℤ} (f0 : f ≠ 0) (fa : Polynomial.eval α (Polynomial.map (algebraMap ℤ ℝ) f) = 0) : ∃ (A : ℝ), 0 < A ∧ ∀ (a : ℤ) (b : ℕ), 1 ≤ (↑b + 1) ^ f.natDegree * (|α - ↑a / (↑b + 1)| * A)

theorem Liouville.transcendental {x : ℝ} (lx : Liouville x) : Transcendental ℤ x

Liouville's Theorem