Lemma 2.23.
Assuming that:
m
∈
ℕ
φ
:
[
m
]
→
ℤ
∕
p
ℤ
be given by
x
↦
t
x
for some
t
≠
0
𝜀
>
0
Then
there exists a partition of
[
m
]
into progressions
P
i
of length
l
i
∈
[
𝜀
m
2
,
𝜀
m
]
such that
diam
(
φ
(
P
i
)
)
=
max
x
,
y
∈
P
i
|
φ
(
x
)
−
φ
(
y
)
|
≤
𝜀
p
for all
i
.