Admissible absolute values

This file defines a structure AbsoluteValue.IsAdmissible which we use to show the class number of the ring of integers of a global field is finite.

Main definitions

• AbsoluteValue.IsAdmissible abv states the absolute value abv : R → ℤ respects the Euclidean domain structure on R, and that a large enough set of elements of R^n contains a pair of elements whose remainders are pointwise close together.

Main results

• AbsoluteValue.absIsAdmissible shows the "standard" absolute value on ℤ, mapping negative x to -x, is admissible.
• Polynomial.cardPowDegreeIsAdmissible shows cardPowDegree, mapping p : Polynomial 𝔽_q to q ^ degree p, is admissible

structure AbsoluteValue.IsAdmissible {R : Type u_1} [EuclideanDomain R] (abv : AbsoluteValue R ℤ) extends AbsoluteValue.IsEuclidean : Type

An absolute value R → ℤ is admissible if it respects the Euclidean domain structure and a large enough set of elements in R^n will contain a pair of elements whose remainders are pointwise close together.

• map_lt_map_iff' : ∀ {x y : R}, abv x < abv y ↔ EuclideanDomain.r x y
• card : ℝ → ℕ
• exists_partition' : ∀ (n : ℕ) {ε : ℝ}, 0 < ε → ∀ {b : R}, b ≠ 0 → ∀ (A : Fin n → R), ∃ (t : Fin n → Fin (self.card ε)), ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ → ↑(abv (A i₁ % b - A i₀ % b)) < abv b • ε

  For all ε > 0 and finite families A, we can partition the remainders of A mod b into abv.card ε sets, such that all elements in each part of remainders are close together.

theorem AbsoluteValue.IsAdmissible.exists_partition' {R : Type u_1} [EuclideanDomain R] {abv : AbsoluteValue R ℤ} (self : abv.IsAdmissible) (n : ℕ) {ε : ℝ} : 0 < ε → ∀ {b : R}, b ≠ 0 → ∀ (A : Fin n → R), ∃ (t : Fin n → Fin (self.card ε)), ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ → ↑(abv (A i₁ % b - A i₀ % b)) < abv b • ε

For all ε > 0 and finite families A, we can partition the remainders of A mod b into abv.card ε sets, such that all elements in each part of remainders are close together.

theorem AbsoluteValue.IsAdmissible.exists_partition {R : Type u_1} [EuclideanDomain R] {abv : AbsoluteValue R ℤ} {ι : Type u_2} [Finite ι] {ε : ℝ} (hε : 0 < ε) {b : R} (hb : b ≠ 0) (A : ι → R) (h : abv.IsAdmissible) : ∃ (t : ι → Fin (h.card ε)), ∀ (i₀ i₁ : ι), t i₀ = t i₁ → ↑(abv (A i₁ % b - A i₀ % b)) < abv b • ε

For all ε > 0 and finite families A, we can partition the remainders of A mod b into abv.card ε sets, such that all elements in each part of remainders are close together.

theorem AbsoluteValue.IsAdmissible.exists_approx_aux {R : Type u_1} [EuclideanDomain R] {abv : AbsoluteValue R ℤ} (n : ℕ) (h : abv.IsAdmissible) {ε : ℝ} (_hε : 0 < ε) {b : R} (_hb : b ≠ 0) (A : Fin (h.card ε ^ n).succ → Fin n → R) : ∃ (i₀ : Fin (h.card ε ^ n).succ) (i₁ : Fin (h.card ε ^ n).succ), i₀ ≠ i₁ ∧ ∀ (k : Fin n), ↑(abv (A i₁ k % b - A i₀ k % b)) < abv b • ε

Any large enough family of vectors in R^n has a pair of elements whose remainders are close together, pointwise.

theorem AbsoluteValue.IsAdmissible.exists_approx {R : Type u_1} [EuclideanDomain R] {abv : AbsoluteValue R ℤ} {ι : Type u_2} [Fintype ι] {ε : ℝ} (hε : 0 < ε) {b : R} (hb : b ≠ 0) (h : abv.IsAdmissible) (A : Fin (h.card ε ^ Fintype.card ι).succ → ι → R) : ∃ (i₀ : Fin (h.card ε ^ Fintype.card ι).succ) (i₁ : Fin (h.card ε ^ Fintype.card ι).succ), i₀ ≠ i₁ ∧ ∀ (k : ι), ↑(abv (A i₁ k % b - A i₀ k % b)) < abv b • ε

Any large enough family of vectors in R^ι has a pair of elements whose remainders are close together, pointwise.