# Documentation

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} [] (abv : ) extends :
• map_lt_map_iff' : ∀ {x y : R}, abv x < abv y
• card :
• exists_partition' : ∀ (n : ) {ε : }, 0 < ε∀ {b : R}, b 0∀ (A : Fin nR), t, ∀ (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.

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.

Instances For
theorem AbsoluteValue.IsAdmissible.exists_partition {R : Type u_1} [] {abv : } {ι : Type u_2} [] {ε : } (hε : 0 < ε) {b : R} (hb : b 0) (A : ιR) (h : ) :
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.

theorem AbsoluteValue.IsAdmissible.exists_approx_aux {R : Type u_1} [] {abv : } (n : ) (h : ) {ε : } (_hε : 0 < ε) {b : R} (_hb : b 0) (A : Fin ()Fin nR) :
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.

theorem AbsoluteValue.IsAdmissible.exists_approx {R : Type u_1} [] {abv : } {ι : Type u_2} [] {ε : } (hε : 0 < ε) {b : R} (hb : b 0) (h : ) (A : ιR) :
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.