Let
d
∈
N
d\in \mathbb {N}
and
p
i
p_i
be an integral polynomial with
p
i
(
0
)
=
0
,
1
≤
i
≤
d
p_i(0)=0,1\leq i\leq d
. It is shown that if
S
S
is thickly syndetic in
Z
\mathbb {Z}
, then
{
(
m
,
n
)
∈
Z
2
:
m
+
p
1
(
n
)
,
m
+
p
2
(
n
)
,
…
,
m
+
p
d
(
n
)
∈
S
}
\{(m,n)\in \mathbb {Z}^2:m+p_1(n),m+p_2(n),\ldots ,m+p_d(n)\in S\}
is thickly syndetic in
Z
2
\mathbb {Z}^2
.
Meanwhile, we construct a transitive, strong mixing and non-minimal topological dynamical system
(
X
,
T
)
(X,T)
, such that the set
{
x
∈
X
:
∀
open
U
∋
x
,
∃
n
∈
Z
s.t.
T
n
x
∈
U
,
T
2
n
x
∈
U
}
\{x\in X:\forall \ \text {open}\ U\ni x,\exists \ n\in \mathbb {Z}\ \text {s.t.}\ T^{n}x\in U,T^{2n}x\in U\}
is not dense in
X
X
.