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→ 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

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

• card : ℝ → ℕ

• 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.

  exists_partition' : ∀ (n : ℕ) {ε : ℝ}, 0 < ε → ∀ {b : R}, b ≠ 0 → ∀ (A : Fin n → R), ∃ t, ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ → ↑(↑abv (A i₁ % b - A i₀ % b)) < ↑abv b • ε

An absolute value R → ℤ→ 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.

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theorem AbsoluteValue.IsAdmissible.exists_partition {R : Type u_2} [inst : EuclideanDomain R] {abv : AbsoluteValue R ℤ} {ι : Type u_1} [inst : Fintype ι] {ε : ℝ} (hε : 0 < ε) {b : R} (hb : b ≠ 0) (A : ι → R) (h : AbsoluteValue.IsAdmissible abv) : ∃ t, ∀ (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.

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

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