ℤ-lattices #

Let E be a finite dimensional vector space over a NormedLinearOrderedField K with a solid norm that is also a FloorRing, e.g. ℝ. A (full) ℤ-lattice L of E is a discrete subgroup of E such that L spans E over K.

A ℤ-lattice L can be defined in two ways:

For b a basis of E, then L = Submodule.span ℤ (Set.range b) is a ℤ-lattice of E
As an AddSubgroup E with the additional properties:
- DiscreteTopology L, that is L is discrete
- Submodule.span ℝ (L : Set E) = ⊤, that is L spans E over K.

Results about the first point of view are in the Zspan namespace and results about the second point of view are in the Zlattice namespace.

Main results #

Zspan.isAddFundamentalDomain: for a ℤ-lattice Submodule.span ℤ (Set.range b), proves that the set defined by Zspan.fundamentalDomain is a fundamental domain.
Zlattice.module_free: an AddSubgroup of E that is discrete and spans E over K is a free ℤ-module
Zlattice.rank: an AddSubgroup of E that is discrete and spans E over K is a free ℤ-module of ℤ-rank equal to the K-rank of E

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def Zspan.fundamentalDomain {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) :

Set E

The fundamental domain of the ℤ-lattice spanned by b. See Zspan.isAddFundamentalDomain for the proof that it is a fundamental domain.

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@[simp]

theorem Zspan.mem_fundamentalDomain {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) {m : E} :

m ∈ Zspan.fundamentalDomain b ↔ ∀ (i : ι), ↑(↑b.repr m) i ∈ Set.Ico 0 1

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theorem Zspan.map_fundamentalDomain {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) {F : Type u_4} [NormedAddCommGroup F] [NormedSpace K F] (f : E ≃ₗ[K] F) :

↑f '' Zspan.fundamentalDomain b = Zspan.fundamentalDomain (Basis.map b f)

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@[simp]

theorem Zspan.fundamentalDomain_reindex {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) {ι' : Type u_4} (e : ι ≃ ι') :

Zspan.fundamentalDomain (Basis.reindex b e) = Zspan.fundamentalDomain b

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theorem Zspan.fundamentalDomain_pi_basisFun {ι : Type u_2} [Fintype ι] :

Zspan.fundamentalDomain (Pi.basisFun ℝ ι) = Set.pi Set.univ fun x => Set.Ico 0 1

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def Zspan.floor {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) :

{ x // x ∈ Submodule.span ℤ (Set.range ↑b) }

The map that sends a vector of E to the element of the ℤ-lattice spanned by b obtained by rounding down its coordinates on the basis b.

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def Zspan.ceil {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) :

{ x // x ∈ Submodule.span ℤ (Set.range ↑b) }

The map that sends a vector of E to the element of the ℤ-lattice spanned by b obtained by rounding up its coordinates on the basis b.

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@[simp]

theorem Zspan.repr_floor_apply {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) (i : ι) :

↑(↑b.repr ↑(Zspan.floor b m)) i = ↑⌊↑(↑b.repr m) i⌋

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@[simp]

theorem Zspan.repr_ceil_apply {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) (i : ι) :

↑(↑b.repr ↑(Zspan.ceil b m)) i = ↑⌈↑(↑b.repr m) i⌉

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@[simp]

theorem Zspan.floor_eq_self_of_mem {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) (h : m ∈ Submodule.span ℤ (Set.range ↑b)) :

↑(Zspan.floor b m) = m

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@[simp]

theorem Zspan.ceil_eq_self_of_mem {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) (h : m ∈ Submodule.span ℤ (Set.range ↑b)) :

↑(Zspan.ceil b m) = m

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def Zspan.fract {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) :

The map that sends a vector E to the fundamentalDomain of the lattice, see Zspan.fract_mem_fundamentalDomain, and fractRestrict for the map with the codomain restricted to fundamentalDomain.

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theorem Zspan.fract_apply {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) :

Zspan.fract b m = m - ↑(Zspan.floor b m)

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@[simp]

theorem Zspan.repr_fract_apply {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) (i : ι) :

↑(↑b.repr (Zspan.fract b m)) i = Int.fract (↑(↑b.repr m) i)

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@[simp]

theorem Zspan.fract_fract {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) :

Zspan.fract b (Zspan.fract b m) = Zspan.fract b m

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@[simp]

theorem Zspan.fract_zspan_add {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) {v : E} (h : v ∈ Submodule.span ℤ (Set.range ↑b)) :

Zspan.fract b (v + m) = Zspan.fract b m

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@[simp]

theorem Zspan.fract_add_zspan {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) {v : E} (h : v ∈ Submodule.span ℤ (Set.range ↑b)) :

Zspan.fract b (m + v) = Zspan.fract b m

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theorem Zspan.fract_eq_self {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] {b : Basis ι K E} [FloorRing K] [Fintype ι] {x : E} :

Zspan.fract b x = x ↔ x ∈ Zspan.fundamentalDomain b

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theorem Zspan.fract_mem_fundamentalDomain {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (x : E) :

Zspan.fract b x ∈ Zspan.fundamentalDomain b

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def Zspan.fractRestrict {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (x : E) :

↑(Zspan.fundamentalDomain b)

The map fract with codomain restricted to fundamentalDomain.

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theorem Zspan.fractRestrict_surjective {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] :

Function.Surjective (Zspan.fractRestrict b)

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@[simp]

theorem Zspan.fractRestrict_apply {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (x : E) :

↑(Zspan.fractRestrict b x) = Zspan.fract b x

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theorem Zspan.fract_eq_fract {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (m : E) (n : E) :

Zspan.fract b m = Zspan.fract b n ↔ -m + n ∈ Submodule.span ℤ (Set.range ↑b)

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theorem Zspan.norm_fract_le {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] [HasSolidNorm K] (m : E) :

‖Zspan.fract b m‖ ≤ Finset.sum Finset.univ fun i => ‖↑b i‖

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@[simp]

theorem Zspan.coe_floor_self {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [FloorRing K] [Fintype ι] [Unique ι] (k : K) :

↑(Zspan.floor (Basis.singleton ι K) k) = ↑⌊k⌋

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@[simp]

theorem Zspan.coe_fract_self {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [FloorRing K] [Fintype ι] [Unique ι] (k : K) :

Zspan.fract (Basis.singleton ι K) k = Int.fract k

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theorem Zspan.fundamentalDomain_isBounded {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Finite ι] [HasSolidNorm K] :

Bornology.IsBounded (Zspan.fundamentalDomain b)

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theorem Zspan.vadd_mem_fundamentalDomain {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (y : { x // x ∈ Submodule.span ℤ (Set.range ↑b) }) (x : E) :

y +ᵥ x ∈ Zspan.fundamentalDomain b ↔ y = -Zspan.floor b x

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theorem Zspan.exist_unique_vadd_mem_fundamentalDomain {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Finite ι] (x : E) :

∃! v, v +ᵥ x ∈ Zspan.fundamentalDomain b

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def Zspan.quotientEquiv {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] :

E ⧸ Submodule.span ℤ (Set.range ↑b) ≃ ↑(Zspan.fundamentalDomain b)

The map Zspan.fractRestrict defines an equiv map between E ⧸ span ℤ (Set.range b) and Zspan.fundamentalDomain b.

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@[simp]

theorem Zspan.quotientEquiv_apply_mk {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (x : E) :

↑(Zspan.quotientEquiv b) (Submodule.Quotient.mk x) = Zspan.fractRestrict b x

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@[simp]

theorem Zspan.quotientEquiv.symm_apply {E : Type u_1} {ι : Type u_2} {K : Type u_3} [NormedLinearOrderedField K] [NormedAddCommGroup E] [NormedSpace K E] (b : Basis ι K E) [FloorRing K] [Fintype ι] (x : ↑(Zspan.fundamentalDomain b)) :

↑(Zspan.quotientEquiv b).symm x = Submodule.Quotient.mk ↑x

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theorem Zspan.fundamentalDomain_measurableSet {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (b : Basis ι ℝ E) [MeasurableSpace E] [OpensMeasurableSpace E] [Finite ι] :

MeasurableSet (Zspan.fundamentalDomain b)

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theorem Zspan.isAddFundamentalDomain {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (b : Basis ι ℝ E) [Finite ι] [MeasurableSpace E] [OpensMeasurableSpace E] (μ : MeasureTheory.Measure E) :

MeasureTheory.IsAddFundamentalDomain { x // x ∈ Submodule.toAddSubgroup (Submodule.span ℤ (Set.range ↑b)) } (Zspan.fundamentalDomain b)

For a ℤ-lattice Submodule.span ℤ (Set.range b), proves that the set defined by Zspan.fundamentalDomain is a fundamental domain.

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theorem Zspan.measure_fundamentalDomain {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (b : Basis ι ℝ E) [Fintype ι] [DecidableEq ι] [MeasurableSpace E] (μ : MeasureTheory.Measure E) [BorelSpace E] [MeasureTheory.Measure.IsAddHaarMeasure μ] (b₀ : Basis ι ℝ E) :

↑↑μ (Zspan.fundamentalDomain b) = ENNReal.ofReal |Matrix.det (Basis.toMatrix b₀ ↑b)| * ↑↑μ (Zspan.fundamentalDomain b₀)

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@[simp]

theorem Zspan.volume_fundamentalDomain {ι : Type u_2} [Fintype ι] [DecidableEq ι] (b : Basis ι ℝ (ι → ℝ)) :

↑↑MeasureTheory.volume (Zspan.fundamentalDomain b) = ENNReal.ofReal |Matrix.det (↑Matrix.of ↑b)|

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theorem Zlattice.FG (K : Type u_1) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E] [ProperSpace E] {L : AddSubgroup E} [DiscreteTopology { x // x ∈ L }] (hs : Submodule.span K ↑L = ⊤) :

AddSubgroup.FG L

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theorem Zlattice.module_finite (K : Type u_1) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E] [ProperSpace E] {L : AddSubgroup E} [DiscreteTopology { x // x ∈ L }] (hs : Submodule.span K ↑L = ⊤) :

Module.Finite ℤ { x // x ∈ L }

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theorem Zlattice.module_free (K : Type u_1) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E] [ProperSpace E] {L : AddSubgroup E} [DiscreteTopology { x // x ∈ L }] (hs : Submodule.span K ↑L = ⊤) :

Module.Free ℤ { x // x ∈ L }

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theorem Zlattice.rank (K : Type u_1) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E] [ProperSpace E] {L : AddSubgroup E} [DiscreteTopology { x // x ∈ L }] (hs : Submodule.span K ↑L = ⊤) :

FiniteDimensional.finrank ℤ { x // x ∈ L } = FiniteDimensional.finrank K E