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LeanCamCombi
.
GrowthInGroups
.
Lecture3
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Imports
Init
LeanCamCombi.Mathlib.Combinatorics.Additive.ApproximateSubgroup
Mathlib.Geometry.Group.Growth.QuotientInter
Imported by
GrowthInGroups
.
Lecture3
.
lemma_3_1
GrowthInGroups
.
Lecture3
.
lemma_3_2
GrowthInGroups
.
Lecture3
.
proposition_3_3
GrowthInGroups
.
Lecture3
.
fact_3_5
GrowthInGroups
.
Lecture3
.
proposition_3_6_1
GrowthInGroups
.
Lecture3
.
proposition_3_6_2
GrowthInGroups
.
Lecture3
.
lemma_3_7
GrowthInGroups
.
Lecture3
.
lemma_3_8_1
GrowthInGroups
.
Lecture3
.
lemma_3_8_2
source
theorem
GrowthInGroups
.
Lecture3
.
lemma_3_1
{
G
:
Type
u_1}
[
Group
G
]
{
K
:
ℝ
}
[
DecidableEq
G
]
{
A
:
Finset
G
}
(
hA₁
:
1
∈
A
)
(
hAsymm
:
A
⁻¹
=
A
)
(
hA
:
↑
(
A
^
3
).
card
≤
K
*
↑
A
.
card
)
:
IsApproximateSubgroup
(
K
^
3
) (
↑
A
^
2
)
source
theorem
GrowthInGroups
.
Lecture3
.
lemma_3_2
{
G
:
Type
u_1}
[
Group
G
]
{
K
:
ℝ
}
[
DecidableEq
G
]
{
A
B
:
Finset
G
}
(
hB
:
B
.
Nonempty
)
(
hK
:
↑
(
A
*
B
).
card
≤
K
*
↑
B
.
card
)
:
∃
F
⊆
A
,
↑
F
.
card
≤
K
∧
A
⊆
F
*
(
B
/
B
)
source
theorem
GrowthInGroups
.
Lecture3
.
proposition_3_3
{
G
:
Type
u_1}
[
Group
G
]
{
K
:
ℝ
}
[
DecidableEq
G
]
{
A
:
Finset
G
}
(
hA₀
:
A
.
Nonempty
)
(
hA
:
↑
(
A
^
2
).
card
≤
K
*
↑
A
.
card
)
:
∃
S
⊆ (
A
⁻¹
*
A
)
^
2
,
IsApproximateSubgroup
(
2
^
12
*
K
^
36
)
↑
S
∧
↑
S
.
card
≤
16
*
K
^
12
*
↑
A
.
card
∧
∃
a
∈
A
,
↑
A
.
card
/
(
2
*
K
)
≤
↑
(
A
∩
MulOpposite.op
a
•
S
).
card
source
theorem
GrowthInGroups
.
Lecture3
.
fact_3_5
{
G
:
Type
u_1}
{
H
:
Type
u_2}
[
Group
G
]
[
Group
H
]
{
K
:
ℝ
}
{
A
:
Set
G
}
(
hA
:
IsApproximateSubgroup
K
A
)
(
π
:
G
→*
H
)
:
IsApproximateSubgroup
K
(
⇑
π
''
A
)
source
theorem
GrowthInGroups
.
Lecture3
.
proposition_3_6_1
{
G
:
Type
u_1}
[
Group
G
]
{
A
B
:
Set
G
}
{
K
L
:
ℝ
}
{
m
n
:
ℕ
}
(
hA
:
IsApproximateSubgroup
K
A
)
(
hB
:
IsApproximateSubgroup
L
B
)
(
hm
:
2
≤
m
)
(
hn
:
2
≤
n
)
:
∃ (
F
:
Finset
G
),
↑
F
.
card
≤
K
^
(
m
-
1
)
*
L
^
(
n
-
1
)
∧
A
^
m
∩
B
^
n
⊆
↑
F
*
(
A
^
2
∩
B
^
2
)
source
theorem
GrowthInGroups
.
Lecture3
.
proposition_3_6_2
{
G
:
Type
u_1}
[
Group
G
]
{
A
B
:
Set
G
}
{
K
L
:
ℝ
}
{
m
n
:
ℕ
}
(
hA
:
IsApproximateSubgroup
K
A
)
(
hB
:
IsApproximateSubgroup
L
B
)
(
hm
:
2
≤
m
)
(
hn
:
2
≤
n
)
:
IsApproximateSubgroup
(
K
^
(
2
*
m
-
1
)
*
L
^
(
2
*
n
-
1
)) (
A
^
m
∩
B
^
n
)
source
theorem
GrowthInGroups
.
Lecture3
.
lemma_3_7
{
G
:
Type
u_1}
[
Group
G
]
{
A
B
:
Set
G
}
(
hA
:
A
⁻¹
=
A
)
(
hB
:
B
⁻¹
=
B
)
(
x
y
:
G
)
:
∃ (
z
:
G
),
x
•
A
∩
y
•
B
⊆
z
•
(
A
^
2
∩
B
^
2
)
source
theorem
GrowthInGroups
.
Lecture3
.
lemma_3_8_1
{
G
:
Type
u_1}
[
Group
G
]
{
m
n
:
ℕ
}
{
H
:
Subgroup
G
}
[
H
.
Normal
]
{
A
:
Finset
G
}
:
(
Finset.image
(⇑
(
QuotientGroup.mk'
H
)
)
(
A
^
m
))
.
card
*
{
x
∈
A
^
n
|
x
∈
H
}
.
card
≤
(
A
^
(
m
+
n
)).
card
source
theorem
GrowthInGroups
.
Lecture3
.
lemma_3_8_2
{
G
:
Type
u_1}
[
Group
G
]
{
H
:
Subgroup
G
}
[
H
.
Normal
]
{
A
:
Finset
G
}
(
hAsymm
:
A
⁻¹
=
A
)
:
A
.
card
≤
(
Finset.image
(⇑
(
QuotientGroup.mk'
H
)
)
A
)
.
card
*
{
x
∈
A
^
2
|
x
∈
H
}
.
card