# mathlibdocumentation

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

## Main definitions #

• absolute_value.is_admissible 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 #

• absolute_value.abs_is_admissible shows the "standard" absolute value on ℤ, mapping negative x to -x, is admissible.
• polynomial.card_pow_degree_is_admissible shows card_pow_degree, mapping p : polynomial 𝔽_q to q ^ degree p, is admissible
structure absolute_value.is_admissible {R : Type u_1} (abv : ) :
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.

theorem absolute_value.is_admissible.exists_partition {R : Type u_1} {abv : } {ι : Type u_2} [fintype ι] {ε : } (hε : 0 < ε) {b : R} (hb : b 0) (A : ι → R) (h : abv.is_admissible) :
∃ (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 absolute_value.is_admissible.exists_approx_aux {R : Type u_1} {abv : } (n : ) (h : abv.is_admissible) {ε : } (hε : 0 < ε) {b : R} (hb : b 0) (A : fin (h.card ε ^ n).succfin n → R) :
∃ (i₀ 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 absolute_value.is_admissible.exists_approx {R : Type u_1} {abv : } {ι : Type u_2} [fintype ι] {ε : } (hε : 0 < ε) {b : R} (hb : b 0) (h : abv.is_admissible) (A : fin (h.card ε ^ .succι → R) :
∃ (i₀ i₁ : fin (h.card ε ^ .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.