Let
T
T
be a measure-preserving
Z
ℓ
\mathbb {Z}^\ell
-action on the probability space
(
X
,
B
,
μ
)
,
(X,{\mathcal B},\mu ),
let
q
1
,
…
,
q
m
:
R
→
R
ℓ
q_1,\dots ,q_m\colon \mathbb {R}\to \mathbb {R}^\ell
be vector polynomials, and let
f
0
,
…
,
f
m
∈
L
∞
(
X
)
f_0,\dots ,f_m \in L^\infty (X)
. For any
ϵ
>
0
\epsilon > 0
and multicorrelation sequences of the form
α
(
n
)
=
∫
X
f
0
⋅
T
⌊
q
1
(
n
)
⌋
f
1
⋯
T
⌊
q
m
(
n
)
⌋
f
m
d
μ
\alpha (n) =\int _Xf_0\cdot T^{ \lfloor q_1(n) \rfloor }f_1\cdots T^{ \lfloor q_m(n) \rfloor }f_m\;d\mu
we show that there exists a nil- sequence
ψ
\psi
for which
lim
N
−
M
→
∞
1
N
−
M
∑
n
=
M
N
−
1
|
α
(
n
)
−
ψ
(
n
)
|
≤
ϵ
\lim _{N - M \to \infty } \frac {1}{N-M} \sum _{n=M}^{N-1} |\alpha (n) - \psi (n)| \leq \epsilon
and
lim
N
→
∞
1
π
(
N
)
∑
p
∈
P
∩
[
1
,
N
]
|
α
(
p
)
−
ψ
(
p
)
|
≤
ϵ
.
\lim _{N \to \infty } \frac {1}{\pi (N)} \sum _{p \in \mathbb {P}\cap [1,N]} |\alpha (p) - \psi (p)| \leq \epsilon .
This result simultaneously generalizes previous results of Frantzikinakis and the authors.