Abstract
Let
p
p
$p$
be a prime, let
upper S
S
$S$
be a non-empty subset of
double struck upper F Subscript p
F
p
$\mathbb{F}_p$
and let
0 less than epsilon less than or equals 1
0
<
ϵ
≤
1
$0\lt \epsilon \leq 1$
. We show that there exists a constant
upper C equals upper C left parenthesis p comma epsilon right parenthesis
C
=
C
(
p
,
ϵ
)
$C=C(p, \epsilon )$
such that for every positive integer
k
k
$k$
, whenever
phi 1 comma ellipsis comma phi Subscript k Baseline colon double struck upper F Subscript p Superscript n Baseline right arrow double struck upper F Subscript p Baseline
ϕ
1
,
…
,
ϕ
k
:
F
p
n
→
F
p
$\phi _1, \dots , \phi _k: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$
are linear forms and
upper E 1 comma ellipsis comma upper E Subscript k Baseline
E
1
,
…
,
E
k
$E_1, \dots , E_k$
are subsets of
double struck upper F Subscript p
F
p
$\mathbb{F}_p$
, there exist linear forms
psi 1 comma ellipsis comma psi Subscript upper C Baseline colon double struck upper F Subscript p Superscript n Baseline right arrow double struck upper F Subscript p Baseline
ψ
1
,
…
,
ψ
C
:
F
p
n
→
F
p
$\psi _1, \dots , \psi _C\,:\, \mathbb{F}_p^n \rightarrow \mathbb{F}_p$
and subsets
upper F 1 comma ellipsis comma upper F Subscript upper C Baseline
F
1
,
…
,
F
C
$F_1, \dots , F_C$
of
double struck upper F Subscript p
F
p
$\mathbb{F}_p$
such that the set
upper U equals left brace x element of upper S Superscript n Baseline colon psi 1 left parenthesis x right parenthesis element of upper F 1 comma ellipsis comma psi Subscript upper C Baseline left parenthesis x right parenthesis element of upper F Subscript upper C Baseline right brace
U
=
{
x
∈
S
n
:
ψ
1
(
x
)
∈
F
1
,
…
,
ψ
C
(
x
)
∈
F
C
}
$U=\{x \in S^n\,:\, \psi _1(x) \in F_1, \dots , \psi _C(x) \in F_C\}$
is contained inside the set
upper V equals left brace x element of upper S Superscript n Baseline colon phi 1 left parenthesis x right parenthesis element of upper E 1 comma ellipsis comma phi Subscript k Baseline left parenthesis x right parenthesis element of upper E Subscript k Baseline right brace
V
=
{
x
∈
S
n
:
ϕ
1
(
x
)
∈
E
1
,
…
,
ϕ
k
(
x
)
∈
E
k
}
$V=\{x \in S^n\,:\, \phi _1(x) \in E_1, \dots , \phi _k(x) \in E_k\}$
, and the difference
upper V minus upper U
V
∖
U
$V \setminus U$
has density at most
epsilon
ϵ
$\epsilon$
inside
upper S Superscript n
S
n
$S^n$
. We then generalize this result to one where
phi 1 comma ellipsis comma phi Subscript k Baseline
ϕ
1
,
…
,
ϕ
k
$\phi _1, \dots , \phi _k$
are replaced by homomorphisms
upper G Superscript n Baseline right arrow upper H
G
n
→
H
$G^n \to H$
for some pair of finite Abelian groups
upper G
G
$G$
and
upper H
H
$H$
, and to another where they are replaced by polynomial maps
double struck upper F Subscript p Superscript n Baseline right arrow double struck upper F Subscript p
F
p
n
→
F
p
$\mathbb{F}_p^n \to \mathbb{F}_p$
of small degree.