1 Introduction
Let
$\Omega (n)$
denote the number of prime factors of a natural number n counted with multiplicities. In multiplicative number theory, a central topic is to study the asymptotic distribution of the values of
$\Omega (n)$
.
In 2022, Bergelson and Richter [Reference Bergelson and Richter2] gave an asymptotic characterization of
$\Omega (n)$
from a dynamical point of view. By topological dynamical system, we mean a pair
$(X,T)$
, where X is a compact metric space and
$T:X\to X$
is a homeomorphism. We say that
$(X,T)$
is uniquely ergodic if there is only one T-invariant Borel probability measure on X.
Theorem 1.1. [Reference Bergelson and Richter2, Theorem A]
Let
$(X,T)$
be a uniquely ergodic topological dynamical system with the unique T-invariant Borel probability measure
$\mu $
. Then, for any
${f\in C(X),x\in X}$
,
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(T^{\Omega(n)}x)=\int_{X}f\,d\mu.\end{align*} $$
Later, Loyd [Reference Loyd19] built an analogue of Theorem 1.1 in the sense of norm convergence. By measure-preserving system, we mean a tuple
$(X,\mathcal {X},\mu ,T)$
, where
$(X,\mathcal {X},\mu )$
is a Lebesgue space (for its definition, see [Reference Glasner14, Definition 2.12]) and
$T:X\to X$
is an invertible measure-preserving transformation. We say that
$(X,\mathcal {X},\mu ,T)$
is ergodic if for any
$A\in \mathcal {X}$
with
$\mu (A\Delta T^{-1}A)=0$
, then
$\mu (A)=0$
or
$1$
.
Theorem 1.2. [Reference Loyd19, Theorem 2.5]
Let
$(X,\mathcal {X},\mu ,T)$
be an ergodic measure-preserving system. Then, for any
$f\in L^{2}(\mu )$
,
$$ \begin{align*}\lim_{N\to\infty}\bigg\Vert{\frac{1}{N}\sum_{n=1}^{N}f(T^{\Omega(n)}x)-\int_{X}f\,d\mu}\bigg\Vert_{L^{2}(\mu)}=0.\end{align*} $$
In 2024, Charamaras [Reference Charamaras5] extended Loyd’s result to the double ergodic averages case.
Theorem 1.3. [Reference Charamaras5, Corollary 1.33]
Let T and S be two invertible measure-preserving transformations acting on the Lebesgue space
$(X,\mathcal {X},\mu )$
such that
$(X,\mathcal {X},\mu ,T)$
and
$(X,\mathcal {X},\mu ,S)$
are ergodic. Then, for any
$f,g\in L^{2}(\mu )$
,
$$ \begin{align*}\lim_{N\to\infty}\bigg\Vert{\frac{1}{N}\sum_{n=1}^{N}f(T^{n}x)g(S^{\Omega(n)}x)-\int_{X}f\,d\mu\int_{X}g\,d\mu}\bigg\Vert_{L^{2}(\mu)}=0.\end{align*} $$
For any
$A\subset \mathbb Z^{k}$
, we define
$d^{*}(A)$
by letting
$$ \begin{align*}d^{*}(A)=\sup_{{\Phi}}\limsup_{N\to\infty}\frac{|A\cap \Phi_N|}{|\Phi_N|},\end{align*} $$
where the supremum is taken over all Følner sequences
${\Phi }=\{\Phi _N\}_{N\in \mathbb {N}}$
in
$\mathbb Z^k$
. (A Følner sequence of
$\mathbb Z^k$
is a sequence
$\{\Phi _N\}_{N\in \mathbb {N}}$
of non-empty finite subsets of
$\mathbb Z^k$
such that for each
$h\in \mathbb Z^k$
,
$ \lim _{N\to \infty }({|(\Phi _N+h)\Delta \Phi _N|}/{|\Phi _N|})=0$
.) If
$d^{*} (A)>0$
, we say that A has positive upper Banach density. As a combinatorial application of Theorem 1.3, Charamaras [Reference Charamaras5, Corollary 1.37] showed that for any
$E\subset \mathbb {N}$
with positive upper Banach density, there exist
$m,n\in \mathbb {N}$
such that
$m,m+n,m+\Omega (n)\in E$
.
Motivated by the above results, in this paper, we consider the following ergodic averages:
$$ \begin{align} \frac{1}{N}\sum_{n=1}^{N}w(n)\prod_{i=1}^{k}f_{i}(T_{i}^{P_{i}(n)}x)\cdot f_{k+1}(S^{\Omega(n)}x), \end{align} $$
where
$S,T_1,\ldots ,T_k$
is a family of invertible measure-preserving transformations acting on the Lebesgue space
$(X,\mathcal {X},\mu )$
,
$P_1,\ldots ,P_k\in \mathbb Z[n]$
,
$f_1,\ldots ,f_{k+1}\in L^{\infty }(\mu )$
, and
${w:\mathbb {N}\to \mathbb C}$
is a sequence.
When
$T_1,\ldots ,T_k$
generate a nilpotent group, and
$f_{k+1}$
and w are constant, the norm convergence of (1.1) was proved by Walsh in [Reference Walsh20].
First, we extend Theorem 1.1 to a weighted form.
Theorem 1.4. Let
$(X,\mathcal {X},\mu ,T)$
be a measure-preserving system. Then, for any
${f\in L^{1}(\mu )}$
, there is a full measure subset
$X_{f}$
of X such that for any
$x\in X_{f}$
, any uniquely ergodic topological dynamical system
$(Y,S)$
with the unique S-invariant Borel probability measure
$\nu $
, any
$g\in C(Y)$
and any
$y\in Y$
,
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(T^{n}x)g(S^{\Omega(n)}y)=f^{*}(x)\int_{Y}g\,d\nu,\end{align*} $$
where
$f^{*}$
is the conditional expectation of f with respect to the sub-
$\sigma $
-algebra
$\mathcal {I}(T)$
of
$\mathcal {X}$
generated by all T-invariant sets (for the definition of conditional expectation, see §2.2).
Remark 1.5.
-
(a) Let
$\mathbb {P}$
be the set of all prime numbers. Given
$k{\kern-1pt}\in{\kern-1pt} \mathbb {N}$
, let
${P_1,\ldots ,P_k\in \mathbb Z[n]}$
. Assume that
$(X,\mathcal {X},\mu ,T)$
satisfies the following property: there is
$\mathcal {P}\subset \mathbb {P}$
with positive relative density such that for any distinct
$p,q\in \mathcal {P}\cup \{1\}$
and any
$g_1,\ldots ,g_{2k}\in L^{\infty }(\mu )$
, the limit exists for
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\prod_{i=1}^{k}T^{P_{i}(pn)}g_i\cdot \prod_{j=1}^{k}T^{P_{j}(qn)}g_{k+j}\end{align*} $$
$\mu $
-almost every (a.e.)
$x\in X$
. (We say that
$\mathcal {P}\subset \mathbb {P}$
has positive relative density if
$ \lim _{N\to \infty }({|\{1\le n\le N:n\in \mathcal {P}\}|}/|\{1\le n\le N: n\in \mathbb {P}\}|)>0.$
) Then, by the method used in the proof of Theorem 1.4, and Theorems 2.1, 2.2, 5.1, we have that for any
$f_1,\ldots ,f_{k}\in L^{\infty }(\mu )$
, there are a full measure subset
$X_{f_1,\ldots ,f_k}$
of X and
$f^{*}\in L^{\infty }(\mu )$
such that for any
$x\in X_{f_1,\ldots ,f_k}$
, any uniquely ergodic topological dynamical system
$(Y,S)$
with the unique S-invariant Borel probability measure
$\nu $
, any
$g\in C(Y)$
and any
$y\in Y$
,
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\prod_{i=1}^{k}f_{i}(T^{P_{i}(n)}x)\cdot g(S^{\Omega(n)}y)=f^{*}(x)\int_{Y}g\,d\nu.\end{align*} $$
-
(b) By [Reference Loyd19, Theorem 1.2], in Theorem 1.4, when there is no continuous restriction for g, it may fail even for
$\nu $
-a.e.
$y\in Y$
.
Next, we introduce some applications of Theorem 1.4. For any
$x\in \mathbb R$
,
$[x]$
is the largest integer such that
$0\le x-[x]<1$
. When n is a non-positive integer, we set
$\Omega (n)=0$
. After applying Theorem 1.4 to rotations on tori, we can get the following corollary.
Corollary 1.6. Let
$\alpha>0,\beta \in \mathbb R$
. Let
$(Y,S)$
be a uniquely ergodic topological dynamical system with the unique S-invariant Borel probability measure
$\nu $
. Then, for any
${g\in C(Y)}$
and any
$y\in Y$
,
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}g(S^{\Omega([\alpha n +\beta])}y)=\int_{Y}g\,d\nu.\end{align*} $$
Remark 1.7.
-
(a) Let
$k\in \mathbb {N}$
and
$0\le r<k$
. By applying Corollary 1.6 to
$(\mathbb Z_{k}, S)$
,
$1_{\{r\}}$
and
$y=0$
, where
$S:x\mapsto x+1$
,
$$ \begin{align*}\lim_{N\to\infty}\frac{|\{1\le n\le N:k|(\Omega([\alpha n +\beta])-r)\}|}{N}=\frac{1}{k}.\end{align*} $$
-
(b) By applying Corollary 1.6 to
$(\mathbb Z_{2}, S)$
,
$g:\mathbb Z_{2}\to \{1,-1\},0\mapsto 1,1\mapsto -1$
and
${y=0}$
, where
$S:x\mapsto x+1$
, where
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\mathbf{\unicode{x3bb}}([\alpha n +\beta])=0,\end{align*} $$
$\mathbf {\unicode{x3bb} }$
is the Liouville function, that is,
$\mathbf {\unicode{x3bb} }:\mathbb {N}\to \{1,-1\},n\mapsto (-1)^{\Omega (n)}$
.
We go on to apply Theorem 1.4 to unipotent affine transformations (see [Reference Furstenberg13, pp. 67–69]) to get the following weighted ergodic theorem, which can be viewed as an analogue of Theorem 2.1.
Corollary 1.8. Let
$k\in \mathbb {N}$
and
$\alpha \in \mathbb R\backslash \mathbb Q$
. Let m be the Haar measure on
$\mathbb {T}$
. Then, for any
$f\in L^{1}(m)$
, there is
$A_{f}\subset \mathbb R^{k}$
with zero Lebesgue measure such that for any
$(x_0,\ldots ,x_{k-1})\notin A_{f}$
, any uniquely ergodic topological dynamical system
$(Y,S)$
with the unique S-invariant Borel probability measure
$\nu $
, any
$g\in C(Y)$
and any
$y\in Y$
,
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(\alpha n^{k}+x_{k-1}n^{k-1}+\cdots+x_{1}n+x_0)g(S^{\Omega(n)}y)=\int_{\mathbb{T}}f\,dm\int_{Y}g\,d\nu.\end{align*} $$
Now, let us come back to (1.1). For some of its cases, we can build related mean ergodic theorems, which can be viewed as extensions of Theorem 1.3.
Theorem 1.9. Given
$k\in \mathbb {N}$
, let
$P_1,\ldots ,P_k$
be pairwise independent polynomials with integer coefficients and any one of them is not of the form
$cn^d+b$
. (
$P_1,\ldots ,P_k$
are pairwise independent if for any
$1\le i<j\le k$
, there are no non-zero
$x,y\in \mathbb Z$
such that
$xP_i+yP_j$
is a constant.) Let
$S,T_1,\ldots ,T_{k}$
be a family of invertible measure-preserving transformations acting on the Lebesgue space
$(X,\mathcal {X},\mu )$
. Assume that
$T_1,\ldots ,T_k$
are commuting. Then, for any
$f_1,\ldots ,f_{k+1}\in L^{\infty }(\mu )$
, the limit
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\prod_{i=1}^{k}f_{i}(T_{i}^{P_{i}(n)}x)\cdot f_{k+1}(S^{\Omega(n)}x)\end{align*} $$
exists in
$L^{2}(\mu )$
.
The reason why we make some restrictions on the polynomials
$P_1,\ldots ,P_k$
is to use the pronilfactors to control the
$L^{2}$
-norm of (1.1). If
$T_1=\cdots =T_k$
, then we can leave out the restrictions for polynomials.
Theorem 1.10. Given
$k\in \mathbb {N}$
, let
$P_1,\ldots ,P_k\in \mathbb Z[n]$
. Let
$T,S$
be two invertible measure-preserving transformations acting on the Lebesgue space
$(X,\mathcal {X},\mu )$
. Then, for any
$f_1,\ldots , f_{k+1}\in L^{\infty }(\mu )$
, the limit
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\prod_{i=1}^{k}f_{i}(T^{P_{i}(n)}x)\cdot f_{k+1}(S^{\Omega(n)}x)\end{align*} $$
exists in
$L^{2}(\mu )$
.
Lastly, we apply Theorems 1.9 and 1.10 to search for some additive structures in the sets with positive upper Banach density. Finally, we can get the following results.
Proposition 1.11. Given
$k\in \mathbb {N}$
, let
$P_1,\ldots ,P_k$
be pairwise independent polynomials with integer coefficients and zero constant terms and any one of them is not of the form
$cn^d$
. Then, for every
$A\subset \mathbb {N}^{k+1}$
with positive upper Banach density, there are
$a\in \mathbb {N}^{k+1},d\in \mathbb {N}$
such that
$$ \begin{align*}a,a+P_{1}(d)\vec{e}_{1},\ldots,a+P_{k}(d)\vec{e}_{k},a+\Omega(d)\vec{e}_{k+1}\in A,\end{align*} $$
where
$\{\vec {e}_1,\ldots ,\vec {e}_{k+1}\}$
is the standard basis of
$\mathbb R^{k+1}$
.
Proposition 1.12. Given
$k\in \mathbb {N}$
, let
$P_1,\ldots ,P_k\in \mathbb Z[n]$
with zero constant terms, then for every
$A\subset \mathbb {N}^{2}$
with positive upper Banach density, there are
$(x,y)\in \mathbb {N}^{2},d\in \mathbb {N}$
such that
$$ \begin{align*}(x,y),(x+P_{1}(d),y),\ldots,(x+P_{k}(d),y),(x,y+\Omega(d))\in A.\end{align*} $$
Proposition 1.13. Given
$k\in \mathbb {N}$
, let
$P_1,\ldots ,P_k\in \mathbb Z[n]$
with zero constant terms, then for every
$A\subset \mathbb {N}$
with positive upper Banach density, there are
$a,d\in \mathbb {N}$
such that
$$ \begin{align*}a,a+P_{1}(d),\ldots,a+P_{k}(d),a+\Omega(d)\in A.\end{align*} $$
Conventionally, to prove the above results, we use Furstenberg’s correspondence principle to transfer them into some multiple recurrence results, which can be deduced from Theorems 1.9, 1.10 and the polynomial Szemerédi theorem [Reference Bergelson and Leibman1, Theorem A].
In fact, by our method and [Reference Bergelson and Richter2, Theorem B and Corollary 1.27], Omega function in all results built by us in the paper can be replaced with any completely additive function
$a:\mathbb {N}\to \mathbb {N}\cup \{0\}$
satisfying the following property:
For any uniquely ergodic additive topological dynamical system
$(X,T)$
with the unique T-invariant Borel probability measure
$\mu $
,
$(X,(T^{a(n)})_{n\in \mathbb {N}})$
is an aperiodic, finitely generated and strongly uniquely ergodic multiplicative topological dynamical system with the unique Borel probability measure
$\mu $
that pretends to be invariant under
$(T^{a(n)})_{n\in \mathbb {N}}$
. (For the definitions of aperiodic, finitely generated and strongly uniquely ergodic multiplicative topological dynamical system and Borel probability measures that pretend to be invariant, see [Reference Bergelson and Richter2, Definitions 1.11, 1.13, 1.22].)
2 Preliminaries
2.1 Isomorphism and factors
We say that measure-preserving systems
$(X,\mathcal {X},\mu ,T)$
and
$(Y,\mathcal {Y},\nu ,S)$
are isomorphic if there exists an invertible measure-preserving transformation
$\Phi :X_0\to Y_0$
with
$\Phi \circ T=S\circ \Phi $
, where
$X_0$
is a T-invariant full measure subset of X and
$Y_0$
is an S-invariant full measure subset of Y.
A factor of a measure-preserving system
$(X,\mathcal {X},\mu ,T)$
is a T-invariant sub-
$\sigma $
-algebra of
$\mathcal {X}$
. A factor map from
$(X,\mathcal {X},\mu ,T)$
to
$(Y,\mathcal {Y},\nu ,S)$
is a measurable map
$\pi :X_{0}\rightarrow Y_{0}$
with
$\pi \circ T=S\circ \pi $
and such that
$\nu $
is the image of
$\mu $
under
$\pi $
, where
$X_0$
is a T-invariant full measurable subset of X and
$Y_0$
is an S-invariant full measurable subset of Y. In this case,
$\pi ^{-1}(\mathcal {Y})$
is a factor of
$(X,\mathcal {X},\mu ,T)$
and every factor of
$(X,\mathcal {X},\mu ,T)$
can be obtained in this way.
2.2 Conditional expectation and ergodic decomposition
Given a Lebesgue space
$(X,\mathcal {X},\mu )$
, let
$\mathcal {Y}$
be a sub-
$\sigma $
-algebra of
$\mathcal {X}$
. For any
$f\in L^{1}(X,\mathcal {X},\mu )$
, the conditional expectation of f with respect to
$\mathcal {Y}$
is the function
$\mathbb E_{\mu }(f|\mathcal {Y})$
, defined in
$L^{1}(X,\mathcal {Y},\mu )$
, such that for any
$A\in \mathcal {Y}$
,
$\int _{A}f\,d\mu =\int _{A}\mathbb E_{\mu }(f|\mathcal {Y})\,d\mu $
. Then, there exists a unique
$\mathcal {Y}$
-measurable map
$X\to \mathcal {M}(X,\mathcal {X}),x\mapsto \mu _x$
, called the disintegration of
$\mu $
with respect to
$\mathcal {Y}$
, up to
$\mu $
-null sets such that for any
$f\in L^{\infty }(X,\mathcal {X},\mu )$
,
$\mathbb E_{\mu }(f|\mathcal {Y})(x)=\int _{X}f\,d\mu _{x}$
for
$\mu $
-a.e.
$x\in X$
, where
$\mathcal {M}(X,\mathcal {X})$
is the collection of probability measures on
$(X,\mathcal {X})$
, endowed with the standard Borel structure.
Let
$(X,\mathcal {X},\mu ,T)$
be a measure-preserving system and
$\mathcal {I}(T)$
be the sub-
$\sigma $
-algebra of
$\mathcal {X}$
, generated by all T-invariant sets. The disintegration of
$\mu $
with respect to
$\mathcal {I}(T)$
, denoted by
$X\to \mathcal {M}(X,\mathcal {X}),x\mapsto \mu _x$
, is called the ergodic decomposition of
$\mu $
with respect to T. Then, for
$\mu $
-a.e.
$x\in X$
,
$(X,\mathcal {X},\mu _x,T)$
is an ergodic measure-preserving system.
2.3 Nilsequences
Let
$k\in \mathbb {N}$
. A k-step nilmanifold X is a quotient space
$G/\Gamma $
, where G is a k-step nilpotent Lie group and
$\Gamma $
is a cocompact discrete subgroup of G. A basic k-step nilsequence is the sequence
$\{f(a^{n}\cdot x)\}_{n\in \mathbb Z}$
, where
${f\in C(X),a\in G,x\in X}$
. A k-step nilsequence is a uniform limit of basic k-step nilsequences. Clearly, all k-step nilsequences are an invariant algebra of
$l^{\infty }(\mathbb Z)$
under translation (for more details, see [Reference Host and Kra16, Ch. 11]). By [Reference Host and Kra16, Proposition 11.13], we have that for any nilsequence
$\{b_n\}_{n\in \mathbb Z}$
, the limit
$ \lim _{N\to \infty }({1}/{N})\sum _{n=1}^{N}b_{n}$
exists.
For nilsequences, Bergelson and Richter [Reference Bergelson and Richter2] built a disjointness result on it.
Theorem 2.1. [Reference Bergelson and Richter2, Corollary 1.27 and Lemma 6.3]
Let
$(X,T)$
be a uniquely ergodic topological dynamical system with the unique T-invariant Borel probability measure
$\mu $
. Let
$\{b_n\}_{n\in \mathbb Z}$
be a nilsequence. Then, for any
$f\in C(X),x\in X$
,
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}b_{n}f(T^{\Omega(n)}x)=\int_{X}f\,d\mu\cdot \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}b_{n}.\end{align*} $$
2.4 Pronilfactors
Fix a measure-preserving system
$(X,\mathcal {X},\mu ,T)$
. For any
${f\in L^{\infty }(\mu )}$
, we let 
. Next, we define 
inductively. For each
$k\ge 1$
and any
$f\in L^{\infty }(\mu )$
, we let
By [Reference Host and Kra16, Proposition 8.16], the limit in the above equation always exists. So, 
is well defined for each
$k\in \mathbb {N}$
. By [Reference Host and Kra16, Theorem 9.7], there exists a factor
$\mathcal {Z}_{k}(T)$
of
$(X,\mathcal {X},\mu ,T)$
, called the k-step factor, such that for any
$f\in L^{\infty }(\mu )$
, 
if and only if
$\mathbb {E}(f|\mathcal {Z}_{k}(T))=0$
. By [Reference Host and Kra16, (8.15)], we know that for any
$k\in \mathbb {N}$
,
$\mathcal {Z}_{k}(T)\subset \mathcal {Z}_{k+1}(T)$
. So, we can define the factor
$\mathcal {Z}_{\infty }(T)$
of
$(X,\mathcal {X},\mu ,T)$
, called the
$\infty $
-step factor, by letting it be the smallest
$\sigma $
-algebra containing
$\bigcup _{k= 1}^{\infty }\mathcal {Z}_{k}(T)$
(for more details, see [Reference Host and Kra16, Chs. 8, 9, 16]).
Next, we introduce a structure theorem for general measure-preserving systems, which will be used in the proofs of the main results of this paper.
Theorem 2.2. [Reference Host and Kra16, Theorem 16.10]
Let
$k{\kern-1.2pt}\in{\kern-1.2pt} \mathbb {N}$
and let
$(X,{\kern-1pt}\mathcal {X},{\kern-1pt}\mu,{\kern-1.2pt}T)$
be a measure-preserving system. Then, for all
$p\ge 1$
and
$\epsilon>0$
, every
$f\in L^{\infty }(\mu )$
admits a decomposition
$$ \begin{align*}f=f_{\mathrm{unif}}+f_{\mathrm{nil}}+f_{\mathrm{sml}},\end{align*} $$
where:
-
(i)
; -
(ii) for
$\mu $
-a.e.
$x\in X$
, the sequence
$\{f_{\mathrm{nil}}(T^{n}x)\}_{n\in \mathbb Z}$
is a k-step nilsequence; -
(iii)
$f_{\mathrm{sml}}\in L^{\infty }(\mu )$
satisfies
$\Vert f_{\mathrm{sml}}\Vert _{L^{p}(\mu )}<\epsilon $
.
;
Furthermore,
$\Vert f_{\mathrm{nil}}\Vert _{L^{\infty }(\mu )}\le \Vert f\Vert _{L^{\infty }(\mu )}$
and
$\Vert f_{\mathrm{nil}}+f_{\mathrm{sml}}\Vert _{L^{\infty }(\mu )}\le \Vert f\Vert _{L^{\infty }(\mu )}$
.
2.5 An orthogonality criterion
The following orthogonality criterion (see [Reference Daboussi6, Lemma 1], [Reference Kátai17, (3.1)], [Reference Bourgain, Sarnak and Ziegler4, Theorem 2]) is an application of the Turán–Kubilius inequality.
Lemma 2.3. [Reference Charamaras5, Lemma 2.14]
Let
$\{A(n)\}_{n\ge 1}$
be a bounded sequence in Hilbert space
$\mathcal {H}$
. If
$\mathcal {P}\subset \mathbb {P}$
with positive relative density such that for any distinct
$p,q\in \mathcal {P}$
,
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\langle A(pn),A(qn)\rangle=0,\end{align*} $$
then
$$ \begin{align*}\lim_{N\to\infty}\bigg\Vert{\frac{1}{N}\sum_{n=1}^{N}A(n)}\bigg\Vert=0.\end{align*} $$
3 Proof of Theorem 1.4
First, let us recall Bourgain’s double recurrence theorem.
Theorem 3.1. [Reference Bourgain3, Main Theorem and (2.3)]
Let
$(X,\mathcal {X},\mu ,T)$
be an ergodic measure-preserving system. Fix two distinct non-zero integers a and b. Fix
$f,g\in L^{\infty }(\mu )$
. If f or g is orthogonal to the closed subspace of
$L^{2}(\mu )$
spanned by all eigenfunctions with respect to T, then
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(T^{an}x)g(T^{bn}x)=0\end{align*} $$
for
$\mu $
-a.e.
$x\in X$
.
Now, we prove Theorem 1.4.
Proof of Theorem 1.4
The whole proof is divided into three steps.
Step I. Reduction to the ergodic case. In this step, we show that to verify that Theorem 1.4 holds, it suffices to prove that Theorem 1.4 holds for all ergodic measure-preserving systems.
Suppose that Theorem 1.4 holds for all ergodic measure-preserving systems. Next, we use contradiction argument to verify that Theorem 1.4 holds.
If Theorem 1.4 does not hold, then there is a measure-preserving system
$(X,\mathcal {X},\mu ,T)$
and
$f\in L^{1}(\mu )$
such that there is a measurable set
$X'\subset X$
of positive
$\mu $
-measure such that for each
$x\in X'$
, there are a uniquely ergodic topological dynamical system
$(Y_{x},S_{x})$
with the unique
$S_{x}$
-invariant Borel probability measure
$\nu _{x}$
,
$g_x\in C(Y)$
and
$y_x\in Y$
such that
$$ \begin{align} \limsup_{N\to\infty}\bigg|\frac{1}{N}\sum_{n=1}^{N}f(T^{n}x)g_{x}(S_{x}^{\Omega(n)}y_{x})-f^{*}(x)\int_{Y_{x}}g_{x}\,d\nu_{x}\bigg|>0, \end{align} $$
where
$$ \begin{align*}f^{*}(x)=\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(T^{n}x).\end{align*} $$
Let
$X\to \mathcal {M}(X),z\to \mu _{z}$
be the ergodic decomposition of
$\mu $
with respect to T. Then, there is
$z\in X$
such that the following hold:
-
(1)
$(X,\mathcal {X},\mu _{z},T)$
is ergodic; -
(2)
$f\in L^{1}(\mu _z)$
; -
(3)
$\mu _{z}(X')>0$
.
Then, by the hypothesis and Birkhoff’s ergodic theorem, we know that there is a measurable set
$X"\subset X$
of full
$\mu _{z}$
-measure such that for each
$x\in X"\cap X'$
,
$$ \begin{align} \lim_{N\to\infty}\bigg|\frac{1}{N}\sum_{n=1}^{N}f(T^{n}x)g_{x}(S_{x}^{\Omega(n)}y_{x})-\int_{X}f\,d\mu_{z}\int_{Y_{x}}g_{x}\,d\nu_{x}\bigg|=0 \end{align} $$
and
$ \int _{X}f\,d\mu _{z}=f^{*}(x)$
. Then, (3.2) contradicts with (3.1). So, Theorem 1.4 can be reduced to the ergodic case.
Step II. Reduction to the
$L^{\infty }$
-functions. Based on Step I, we fix an ergodic measure-preserving system
$(X,\mathcal {X},\mu ,T)$
. In this step, we show that to verify that Theorem 1.4 holds for
$(X,\mathcal {X},\mu ,T)$
, it suffices to prove that Theorem 1.4 holds for all elements of
$L^{\infty }(\mu )$
.
Suppose that Theorem 1.4 holds for all elements of
$L^{\infty }(\mu )$
. Fix
$f\in L^{1}(\mu )$
. Then, there is a sequence of functions
$\{f_j\}_{j\ge 1}\subset L^{\infty }(\mu )$
such that the following hold:
-
(1) for each
$j\ge 1$
,
$\Vert f-f_j\Vert _{L^{1}(\mu )}<1/2^{j}$
; -
(2) for each
$j\ge 1$
, there is a full measure subset
$X_{j}$
of X such that Theorem 1.4 holds for
$f_j$
.
Let
$$ \begin{align*} X_0&=\bigcap_{j\ge 1}X_{j}\cap \bigg\{x\in X:\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(T^{n}x)=\int_{X}f\,d\mu\bigg\}\\ &\quad \cap\bigcap_{j\ge 1}\bigg\{x\in X:\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}|f-f_j|(T^{n}x)=\Vert f-f_j\Vert_{L^{1}(\mu)}\bigg\}. \end{align*} $$
By Birkhoff’s ergodic theorem, we have that
$\mu (X_0)=1$
.
Fix
$x\in X_0$
. Fix a uniquely ergodic topological dynamical system
$(Y,S)$
with the unique S-invariant Borel probability measure
$\nu $
,
$g\in C(Y)$
and
$y\in Y$
. Then,
$$ \begin{align*} & \limsup_{N\to\infty}\bigg|\frac{1}{N}\sum_{n=1}^{N}f(T^{n}x)g(S^{\Omega(n)}y)-\int_{X}f\,d\mu\int_{Y}g\,d\nu\bigg| \\&\!\!\quad\le \limsup_{j\to\infty}\limsup_{N\to\infty}\bigg|\frac{1}{N}\!\sum_{n=1}^{N}(f{\kern-1pt}-{\kern-1pt}f_j)(T^{n}x)g(S^{\Omega(n)}y)\bigg|{\kern-1pt}+{\kern-1pt}\limsup_{j\to\infty}\bigg|\!\int_{X}(f{\kern-1pt}-{\kern-1pt}f_j)\,d\mu\!\int_{Y}\!g\,d\nu\bigg| \\&\!\!\quad\le 2\Vert g\Vert_{L^{\infty}(\nu)}\limsup_{j\to\infty}\Vert f-f_j\Vert_{L^{1}(\mu)} \\&\!\!\quad= 0. \end{align*} $$
So, Theorem 1.4 can be reduced to the
$L^{\infty }$
-functions.
Step III. Proving Theorem 1.4 for the ergodic case and the
$L^{\infty }$
-functions. Fix an ergodic measure-preserving system
$(X,\mathcal {X},\mu ,T)$
and
$1$
-bounded
$f\in L^{\infty }(\mu )$
. Let Z be the sub-
$\sigma $
-algebra of
$\mathcal {X}$
generated by all eigenfunctions with respect to T. Then, we can write f as
$\mathbb E_{\mu }(f|Z)+(f-\mathbb E_{\mu }(f|Z))$
.
By repeating the argument in Step II and the linear property of ergodic averages, we know that to prove that for
$\mathbb E_{\mu }(f|Z)$
, Theorem 1.4 holds, it suffices to show that Theorem 1.4 holds for all eigenfunctions with respect to T. Fix a non-constant eigenfunction h with eigenvalue
$\unicode{x3bb} $
. Clearly,
$\unicode{x3bb} \in \mathbb {T}, \unicode{x3bb} \neq 1$
and
$ \int _{X}h\,d\mu =0$
. Fix
$$ \begin{align*}x\in \{y\in X:h(T^{n}y)=\unicode{x3bb}^{n} h(y)\ \text{for each}\ n\ge 1\}\cap \{y\in X:|h(y)|<\infty\}.\end{align*} $$
Fix a uniquely ergodic topological dynamical system
$(Y,S)$
with the unique S-invariant Borel probability measure
$\nu $
,
$g\in C(Y)$
and
$y\in Y$
. Then,
$$ \begin{align*}\frac{1}{N}\sum_{n=1}^{N}h(T^{n}x)g(S^{\Omega(n)}y)=h(x)\frac{1}{N}\sum_{n=1}^{N}\unicode{x3bb}^{n}g(S^{\Omega(n)}y).\end{align*} $$
By [Reference Bergelson and Richter2, Corollary 1.25] and the fact that
$ \int _{X}h\,d\mu =0$
,
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}h(T^{n}x)g(S^{\Omega(n)}y)=\int_{X}h\,d\mu\int_{Y}g\,d\nu.\end{align*} $$
So, for such h, Theorem 1.4 holds.
When h is a constant, by Theorem 1.1, we know that Theorem 1.4 holds. To sum up, Theorem 1.4 holds for
$\mathbb E_{\mu }(f|Z)$
.
Next, we prove that Theorem 1.4 holds for
$\tilde {f}:=f-\mathbb E_{\mu }(f|Z)$
. Clearly,
$\tilde {f}$
is orthogonal to the closed subspace of
$L^{2}(\mu )$
spanned by all eigenfunctions with respect to T and
$\int _{X}{\tilde {f}\,d\mu =0}$
.
Let
$$ \begin{align*} \bar{X}&=\bigg\{x\in X: \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\tilde{f}(T^{n}x)=0\bigg\} \\ & \quad \cap\bigcap_{p,q\in \mathbb{P},\atop p\neq q}\bigg\{x\in X: \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\tilde{f}(T^{pn}x)\bar{\tilde{f}}(T^{qn}x)=0\bigg\}. \end{align*} $$
By Birkhoff’s ergodic theorem and Theorem 3.1,
$\mu (\bar {X})=1$
.
Fix
$x\in \bar {X}$
. Fix a uniquely ergodic topological dynamical system
$(Y,S)$
with the unique S-invariant Borel probability measure
$\nu $
,
$g\in C(Y)$
and
$y\in Y$
. Note that when
$0\le g\le 1$
, g can be written as
$$ \begin{align*}\tfrac{1}{2}((g+i\sqrt{1-g^2})+(g-i\sqrt{1-g^2})).\end{align*} $$
So, by the linear property of ergodic averages, we can assume that
$|g|\equiv 1$
.
Fix two distinct prime numbers p and q. Then,
$$ \begin{align} & \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\tilde{f}(T^{pn}x)g(S^{\Omega(pn)}y)\cdot \bar{\tilde{f}}(T^{qn}x)\bar{g}(S^{\Omega(qn)}y) \nonumber\\ &\quad= \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\tilde{f}(T^{pn}x) \bar{\tilde{f}}(T^{qn}x)(g\cdot \bar{g})(S^{\Omega(n)+1}y) \nonumber\\ &\quad= \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\tilde{f}(T^{pn}x) \bar{\tilde{f}}(T^{qn}x) \nonumber\\ &\quad= 0. \end{align} $$
By combining (3.3), Lemma 2.3 and the fact that
$ \int _{X}\tilde {f}\,d\mu =0$
, we know that
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\tilde{f}(T^{n}x)g(S^{\Omega(n)}y)=\int_{X}\tilde{f}\,d\mu\int_{Y}g\,d\nu.\end{align*} $$
So, Theorem 1.4 holds for
$\tilde {f}$
. This finishes the whole proof.
4 Applications of Theorem 1.4
First, we prove Corollary 1.6.
Proof of Corollary 1.6
Note that for any bounded sequence
$\{a_n\}_{n\ge 1}\subset \mathbb C$
and each
$k\in \mathbb {N}$
,
$$ \begin{align} \lim_{N\to \infty}\bigg|\frac{1}{N}\sum_{n=1}^{N}a_{n}-\frac{1}{N}\sum_{n=1}^{N}a_{n+k}\bigg|=\lim_{N\to \infty}\bigg|\frac{1}{N}\sum_{n=1}^{N}a_{n}-\frac{1}{k}\sum_{i=0}^{k-1}\frac{1}{[N/k]}\sum_{n=1}^{[N/k]}a_{kn+i}\bigg|=0. \end{align} $$
Then, we can assume that
$\beta>0$
and
$\alpha +\beta \ge 1$
.
If
$\alpha \in \mathbb Q$
, then there are
$q,p\in \mathbb {N}$
such that for each
$0\le i<q$
, there are
$d_i\ge 0, 0\le c_i<p$
such that for any
$n\in \mathbb {N}$
,
$[\alpha (qn+i)+\beta ]=p(n+d_i)+c_i$
. By combining (4.1) and [Reference Bergelson and Richter2, Corollary 1.16], we have that Corollary 1.6 holds.
Now, fix
$\alpha \in \mathbb R\backslash \mathbb Q$
. By (4.1), we can assume that
$\alpha>100$
and for each
$n\in \mathbb {N}$
,
$\alpha n+\beta \notin \mathbb {N}$
. For any
$t\in \mathbb R$
, let
$\{t\}=t-[t]$
. Note that
$m\in \{[\alpha n+\beta ]:n\in \mathbb {N}\}$
if and only if
$$ \begin{align*}\bigg\{\frac{m-\beta}{\alpha}\bigg\}\in \bigg(1-\frac{1}{\alpha},1\bigg).\end{align*} $$
Then, there exists an open interval
$A_{\alpha ,\beta }$
of
$\mathbb {T}$
with length
$\alpha ^{-1}$
such that
$m\in \{[\alpha n+\beta ]:n\in \mathbb {N}\}$
if and only if
$m/\alpha \in A_{\alpha ,\beta }$
.
Fix a uniquely ergodic topological dynamical system
$(Y,S)$
with the unique S-invariant Borel probability measure
$\nu $
,
$g\in C(Y)$
and
$y\in Y$
. Let
$T:\mathbb {T}\to \mathbb {T},x\mapsto x+\alpha ^{-1}$
. Then,
$(\mathbb {T},T)$
is uniquely ergodic and the unique T-invariant Borel probability measure is the Haar measure
$\mu $
on
$\mathbb {T}$
. Let
$\mathcal {B}({\mathbb {T}})$
be the Borel
$\sigma $
-algebra on
$\mathbb {T}$
. After applying Theorem 1.4 to
$(\mathbb {T},\mathcal {B}({\mathbb {T}}),\mu ,T)$
and
$1_{A_{\alpha ,\beta }}$
, we have that for any
$\epsilon \in (0,(100\alpha )^{-1})$
, there is
$x_{\epsilon }\in (0,\epsilon )$
such that
$$ \begin{align} \lim_{N\to \infty}\frac{1}{N}\sum_{n=1}^{N}1_{A_{\alpha,\beta}}(x_{\epsilon}+n/\alpha)g(S^{\Omega(n)}y)=\frac{1}{\alpha}\int_{Y}g\,d\nu. \end{align} $$
By Weyl’s uniform distribution theorem,
$$ \begin{align} \lim_{N\to \infty}\frac{1}{N}\sum_{n=1}^{N}|1_{A_{\alpha,\beta}}(x_{\epsilon}+n/\alpha)-1_{A_{\alpha,\beta}}(n/\alpha)|\le 2\epsilon. \end{align} $$
Note that
$$ \begin{align} \frac{1}{N}\sum_{n=1}^{N}g(S^{\Omega([\alpha n+\beta])}y)=\frac{[\alpha N+\beta]}{N}\cdot \frac{1}{[\alpha N+\beta]}\sum_{n=1}^{[\alpha N+\beta]}1_{A_{\alpha,\beta}}(n/\alpha)g(S^{\Omega(n)}y). \end{align} $$
By combining (4.2)–(4.4), we conclude that
$$ \begin{align*}\lim_{N\to \infty}\frac{1}{N}\sum_{n=1}^{N}g(S^{\Omega([\alpha n+\beta])}y)=\int_{Y}g\,d\nu.\end{align*} $$
This finishes the proof.
Next, we show Corollary 1.8.
Proof of Corollary 1.8
For any
$\beta \in \mathbb R$
, we define
$T_{\beta }:\mathbb {T}^{k}\to \mathbb {T}^{k}$
by putting
$$ \begin{align*}T(x_1,\ldots,x_k)=(x_1+\beta,x_2+x_1,\ldots,x_{k}+x_{k-1})\end{align*} $$
for any
$(x_1,\ldots ,x_k)\in \mathbb {T}^{k}$
. By [Reference Einsiedler and Ward7, Corollary 4.22], when
$\beta $
is irrational,
$(\mathbb {T}^{k},T_{\beta })$
is uniquely ergodic and the unique T-invariant Borel probability measure is
$m^{\otimes k}$
, where
$$ \begin{align*}m^{\otimes k}=m\times \cdots \times m\quad(k\ \text{times}).\end{align*} $$
Let
$$ \begin{align*} C=\begin{pmatrix} 1 & 0 & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} &\cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ \binom{k}{0} & \binom{k}{1} & \cdots & \binom{k}{k} \end{pmatrix},\quad B=\begin{pmatrix} 1& 0 &\cdots & 0\\ 1 & 1 & \cdots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & k & \cdots & k^{k} \end{pmatrix}. \end{align*} $$
Let
$\pi _{k}:\mathbb {T}^{k}\to \mathbb {T}$
be the kth coordinate projection. Then, when
$$ \begin{align} (c_0,\ldots,c_k)^{T}=B^{-1}C(x_k,\ldots,x_0)^{T}, \end{align} $$
we have that
$k!c_{k}=x_0$
and for each
$n\in \mathbb {N}$
,
$$ \begin{align} \pi_{k}(T_{x_0}^{n}(x_1,\ldots,x_k))=(c_kn^k+\cdots+c_{1}n+c_0)\,\ \mod 1. \end{align} $$
Fix
$f\in L^{1}(m)$
. Let
$\mathcal {B}({\mathbb {T}^{k}})$
be the Borel
$\sigma $
-algebra on
$\mathbb {T}^{k}$
. After applying Theorem 1.4 to
$(\mathbb {T}^{k},\mathcal {B}({\mathbb {T}^{k}}),m^{\otimes k},T_{k!\alpha })$
and
$f\circ \pi _{k}$
, we know that there is an
$m^{\otimes k}$
-null subset
$X_{f}$
of
$\mathbb {T}^{k}$
such that for any
$(x_1,\ldots ,x_k)\notin X_{f}$
, any uniquely ergodic topological dynamical system
$(Y,S)$
with the unique S-invariant Borel probability measure
$\nu $
, any
$g\in C(Y)$
and any
$y\in Y$
,
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(\pi_{k}(T_{k!\alpha}^{n}(x_1,\ldots,x_k)))g(S^{\Omega(n)}y)=\int_{\mathbb{T}}f\,dm\int_{Y}g\,d\nu.\end{align*} $$
Reference [Reference Folland8, Theorem 2.44.a] tells us that the image of any zero Lebesgue measure subset of
$\mathbb R^{k}$
under an invertible linear map is still of zero Lebesgue measure. Combining this, (4.5) and (4.6), we can find a set
$A_f\subset \mathbb R^{k}$
with zero Lebesgue measure such that for any
$(c_0,\ldots ,c_{k-1})\notin A_{f}$
, any uniquely ergodic topological dynamical system
$(Y,S)$
with the unique S-invariant Borel probability measure
$\nu $
, any
$g\in C(Y)$
and any
$y\in Y$
,
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(\alpha n^k+c_{k-1}n^{k-1}+\cdots+c_{1}n+c_0)g(S^{\Omega(n)}y)=\int_{\mathbb{T}}f\,dm\int_{Y}g\,d\nu.\end{align*} $$
This finishes the proof.
5 Proofs of Theorems 1.9 and 1.10
Before proving Theorems 1.9 and 1.10, let us introduce two results, which point out the characteristic factor behaviour of the
$\infty $
-step factors.
Theorem 5.1. ([Reference Host and Kra15, Theorem 1], [Reference Leibman18, Theorem 3])
Given
$k\in \mathbb {N}$
, let
$P_1,\ldots ,P_k\in \mathbb Z[n]$
. Let T be an invertible measure-preserving transformation acting on the Lebesgue space
$(X,\mathcal {X},\mu )$
. Fix
$f_1,\ldots ,f_{k}\in L^{\infty }(\mu )$
. If there is some
$1\le j\le k$
such that
$\mathbb E_{\mu }(f_j|\mathcal {Z}_{\infty }(T))=0$
, then
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\prod_{i=1}^{k}f_{i}(T^{P_{i}(n)}x)=0\end{align*} $$
in
$L^{2}(\mu )$
.
Theorem 5.2. [Reference Frantzikinakis and Kuca11, Theorem 2.8]
Given
$k\in \mathbb {N}$
, let
$P_1,\ldots ,P_k\in \mathbb Z[n]$
and they are pairwise independent. Let
$T_1,\ldots ,T_{k}$
be a family of commuting invertible measure-preserving transformations acting on the Lebesgue space
$(X,\mathcal {X},\mu )$
. Fix
$f_1,\ldots ,f_{k}\in L^{\infty }(\mu )$
. If there is some
$j\in \{1,\ldots ,k\}$
such that
$\mathbb E_{\mu }(f_j|\mathcal {Z}_{\infty }(T_j))=0$
, then
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\prod_{i=1}^{k}f_{i}(T_{i}^{P_{i}(n)}x)=0\end{align*} $$
in
$L^{2}(\mu )$
.
Now, we begin to prove Theorem 1.9.
Proof of Theorem 1.9
Without loss of generality, we can assume that
$P_{1},\ldots ,P_{k}$
have zero constant terms. Let
$X\to \mathcal {M}(X),x\mapsto \mu _x$
be the ergodic decomposition of
$\mu $
with respect to S. Fix
$1$
-bounded
$g_1,\ldots ,g_{k+1}\in L^{\infty }(\mu )$
. By [Reference Leibman18, Theorem 1], it suffices to prove the following:
$$ \begin{align} & \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}g_{1}(T_{1}^{P_{1}(n)}x)\cdots g_{k}(T_{k}^{P_{k}(n)}x)g_{k+1}(S^{\Omega(n)}x) \nonumber\\& \quad = \int_{X}g_{k+1}\,d\mu_{x}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}g_{1}(T_{1}^{P_{1}(n)}x)\cdots g_{k}(T_{k}^{P_{k}(n)}x) \end{align} $$
in
$L^{2}(\mu )$
.
First, we show that the characteristic factor of the ergodic averages stated in the left side of (5.1) is
$\mathcal {Z}_{\infty }(T_i),1\le i\le k$
.
By [Reference Charamaras5, Proof of Lemma 2.15], we can assume that
$|g_{k+1}|\equiv 1$
. Let the set
$D=\{(p,q)\in \mathbb {P}^{2}:p\neq q\ \text {and there are}\ 1{\kern-1pt}\le{\kern-1pt} i{\kern-1pt}\le{\kern-1pt} j{\kern-1pt}\le{\kern-1pt} k,x,y\in \mathbb Z\backslash \{0\}\ \text {such that}\ xP_{i}(pn)+yP_{j}(qn)\equiv 0\}$
. Then, by the assumption on
$P_1,\ldots ,P_k$
, a simple calculation gives that D is empty or finite.
For each
$n\in \mathbb {N}$
, let
$$ \begin{align*}A(n)=T_{1}^{P_{1}(n)}g_{1}\cdots T_{k}^{P_{k}(n)}g_{k}\cdot S^{\Omega(n)}g_{k+1}.\end{align*} $$
Fix two distinct
$p,q\in \mathbb {P}$
such that
$(p,q)\notin D$
. Then,
$$ \begin{align} & \frac{1}{N}\sum_{n=1}^{N}\langle A(pn),A(qn)\rangle\nonumber\\ &\quad= \frac{1}{N}\sum_{n=1}^{N}\int_{X}\prod_{i=1}^{k}g_{i}(T_{i}^{P_{i}(pn)}x)\cdot \prod_{i=1}^{k}\bar{g}_{i}(T_{i}^{P_{i}(qn)}x)\cdot (g_{k+1}\cdot \bar{g}_{k+1})(S^{\Omega(n)+1}x)\,d\mu(x) \nonumber\\ &\quad= \frac{1}{N}\sum_{n=1}^{N}\int_{X}\prod_{i=1}^{k}g_{i}(T_{i}^{P_{i}(pn)}x)\cdot \prod_{i=1}^{k}\bar{g}_{i}(T_{i}^{P_{i}(qn)}x)\,d\mu(x). \end{align} $$
Note that by the choices of p and q, the polynomial family
$$ \begin{align*}\{P_{1}(pn),\ldots,P_{k}(pn),P_{1}(qn),\ldots,P_{k}(qn)\}\end{align*} $$
is pairwise independent. By Theorem 5.2 and (5.2), we know that if there is some
${1\le j\le k}$
such that
$\mathbb E_{\mu }(g_j|\mathcal {Z}_{\infty }(T_j))=0$
, then
$$ \begin{align} \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\langle A(pn),A(qn)\rangle=0. \end{align} $$
Combining (5.3) and Lemma 2.3, we have that if there is some
$1\le j\le k$
such that
$\mathbb E_{\mu }(g_j|\mathcal {Z}_{\infty }(T_j))=0$
, then
$$ \begin{align} \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}A(n)=0 \end{align} $$
in
$L^{2}(\mu )$
. Equation (5.4) means that
$$ \begin{align} & \lim_{N\to\infty}\bigg|\frac{1}{N}\sum_{n=1}^{N}g_{1}(T_{1}^{P_{1}(n)}x)\cdots g_{k}(T_{k}^{P_{k}(n)}x)g_{k+1}(S^{\Omega(n)}x)\nonumber\\ & \quad -\frac{1}{N}\sum_{n=1}^{N}\mathbb E_{\mu}(g_1|\mathcal{Z}_{\infty}(T_1))(T_{1}^{P_{1}(n)}x)\cdots \mathbb E_{\mu}(g_k|\mathcal{Z}_{\infty}(T_k))(T_{k}^{P_{k}(n)}x)g_{k+1}(S^{\Omega(n)}x)\bigg|=0 \end{align} $$
in
$L^{2}(\mu )$
.
Next, we show that
$$ \begin{align} \begin{aligned} & \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\mathbb E_{\mu}(g_1|\mathcal{Z}_{\infty}(T_1))(T_{1}^{P_{1}(n)}x)\cdots \mathbb E_{\mu}(g_k|\mathcal{Z}_{\infty}(T_k))(T_{k}^{P_{k}(n)}x)g_{k+1}(S^{\Omega(n)}x) \\ &\quad = \int_{X}g_{k+1}d\mu_{x}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\mathbb E_{\mu}(g_1|\mathcal{Z}_{\infty}(T_1))(T_{1}^{P_{1}(n)}x)\cdots \mathbb E_{\mu}(g_k|\mathcal{Z}_{\infty}(T_k))(T_{k}^{P_{k}(n)}x) \end{aligned} \end{align} $$
in
$L^{2}(\mu )$
. By combining Theorem 5.2 and (5.5), we know that if we prove this, then we finish the proof of (5.1).
By Theorem 2.2 and [Reference Host and Kra16, Theorem 14.15], for each
$1\le i\le k$
, there exists a sequence of functions
$\{f_{i,j}\}_{j\ge 1}$
such that the following hold:
-
(1) for each
$1\le i\le k,j\ge 1$
and
$\mu $
-a.e.
$x\in X$
,
$\{f_{i,j}(T_{i}^{P_{i}(n)}x)\}_{n\in \mathbb Z}$
is a nilsequence; -
(2) for each
$1\le i\le k,j\ge 1$
,
$\Vert \mathbb E_{\mu }(g_i|\mathcal {Z}_{\infty }(T_i))-f_{i,j}\Vert _{L^{2k}(\mu )}\le 1/2^{j}$
; -
(3) for each
$1\le i\le k,j\ge 1$
,
$\Vert f_{i,j}\Vert _{L^{\infty }(\mu )}\le 1$
.
Then, there exists
$X_0\in \mathcal {X}$
with
$\mu (X_0)=1$
such that the following hold:
-
(1) for any
$y\in X_0$
, each
$1\le i\le k,j\ge 1$
, and
$\mu _y$
-a.e.
$x\in X$
,
$\{f_{i,j}(T_{i}^{P_{i}(n)}x)\}_{n\in \mathbb Z}$
is a nilsequence; -
(2) for any
$y\in X_0$
,
$(X,\mathcal {X},\mu _y,S)$
is ergodic; -
(3) for any
$y\in X_0$
,
$\Vert g_{k+1}\Vert _{L^{\infty }(\mu _y)}\le 1$
; -
(4) for any
$y\in X_0$
and each
$1\le i\le k,j\ge 1$
,
$\Vert f_{i,j}\Vert _{L^{\infty }(\mu _y)}\le 1$
.
To prove (5.6), it suffices to show that for any
$y\in X_0$
and each
$j\ge 1$
,
$$ \begin{align} & \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f_{1,j}(T_{1}^{P_{1}(n)}x)\cdots f_{k,j}(T_{k}^{P_{k}(n)}x)g_{k+1}(S^{\Omega(n)}x)\nonumber\\& \quad = \int_{X}g_{k+1}\,d\mu_{y}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f_{1,j}(T_{1}^{P_{1}(n)}x)\cdots f_{k,j}(T_{k}^{P_{k}(n)}x) \end{align} $$
in
$L^{2}(\mu _y)$
. To see this, let us do the following calculation:
$$ \begin{align*} & \limsup_{N\to\infty}\bigg\Vert{\frac{1}{N}\sum_{n=1}^{N}\prod_{i=1}^{k}\mathbb E_{\mu}(g_i|\mathcal{Z}_{\infty}(T_i))(T_{i}^{P_{i}(n)}x)\cdot\bigg(g_{k+1}(S^{\Omega(n)}x)-\int_{X}g_{k+1}\,d\mu_{x}\bigg)}\bigg\Vert_{L^{2}(\mu)} \\& \le \limsup_{j\to\infty}\limsup_{N\to\infty}\bigg\Vert{\frac{1}{N}\sum_{n=1}^{N}\prod_{i=1}^{k}f_{i,j}(T_{i}^{P_{i}(n)}x)\cdot\bigg(g_{k+1}(S^{\Omega(n)}x)-\int_{X}g_{k+1}\,d\mu_{x}\bigg)}\bigg\Vert_{L^{2}(\mu)} \\& \quad + 2\limsup_{j\to\infty}\limsup_{N\to\infty}\bigg(\frac{1}{N}\sum_{n=1}^{N}\bigg\Vert{\bigg(\prod_{i=1}^{k}T_{i}^{P_{i}(n)}f_{i,j}-\prod_{i=1}^{k}T_{i}^{P_{i}(n)}\mathbb E_{\mu}(g_i|\mathcal{Z}_{\infty}(T_i))\bigg)}\bigg\Vert_{L^{2}(\mu)}^{2}\bigg)^{1/2} \\& \le \sup_{j\ge 1}\limsup_{N\to\infty}\bigg(\int_{X}\bigg|\frac{1}{N}\sum_{n=1}^{N}\prod_{i=1}^{k}f_{i,j}(T_{i}^{P_{i}(n)}x)\cdot\bigg(\!g_{k+1}(S^{\Omega(n)}x)-\int_{X}g_{k+1}\,d\mu_{x}\!\bigg)\bigg|^{2}\,d\mu(x)\bigg)^{1/2} \\& \le \sup_{j\ge 1}\bigg\Vert{\limsup_{N\to\infty}\bigg\Vert \bigg|\frac{1}{N}\sum_{n=1}^{N}\prod_{i=1}^{k}f_{i,j}(T_{i}^{P_{i}(n)}x)\cdot\bigg(g_{k+1}(S^{\Omega(n)}x)-\int_{X}g_{k+1}\,d\mu_{y}\bigg)\bigg|\bigg\Vert_{L_{x}^{2}(\mu_y)}^{2}}\bigg\Vert_{L_{y}^{1}(\mu)}^{{1}/{2}} \\& = 0, \end{align*} $$
where the last equality comes from (5.7).
Now, we verify (5.7). Fix
$y\in X_0$
and
$j\ge 1$
. Then, we get an ergodic measure-preserving system
$(X,\mathcal {X},\mu _y,S)$
. By [Reference Glasner14, Theorem 15.27], there exists a uniquely ergodic topological dynamical system
$(Y,R)$
with the unique R-invariant Borel probability measure
$\nu $
such that
$(X,\mathcal {X},\mu _y,S)$
and
$(Y,\mathcal B(Y),\nu ,R)$
are isomorphic via the invertible measure-preserving transformation
$\pi :X\to Y$
, where
$\mathcal {B}(Y)$
is the Borel
$\sigma $
-algebra on Y. Then, we can find
$X_1\in \mathcal {X}$
with
$\mu _{y}(X_1)=1$
and a sequence of functions
$\{h_i\}_{i\ge 1}$
in
$C(Y)$
such that the following hold:
-
(1) for each
$i\ge 1$
,
$\Vert h_i\circ \pi - g_{k+1}\Vert _{L^{2}(\mu _y)}\le 1/2^i$
; -
(2) for any
$x\in X_1,1\le t\le k$
,
$\{f_{t,j}(T_{t}^{P_{t}(n)}x)\}_{n\in \mathbb Z}$
is a nilsequence; -
(3) for any
$x\in X_1$
and each
$n\in \mathbb Z$
,
$\pi (S^{n}x)=R^{n}\pi (x)$
; -
(4) for any
$y\in Y$
and each
$i\ge 1$
,
$|h_{i}(y)|\le 1$
.
Note that the product of finitely many nilsequences is still a nilsequence. Then, by Theorem 2.1, we know that for any
$x\in X_1,i\ge 1$
,
$$ \begin{align} & \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f_{1,j}(T_{1}^{P_{1}(n)}x)\cdots f_{k,j}(T_{k}^{P_{k}(n)}x)h_{i}\circ \pi(S^{\Omega(n)}x) \nonumber\\ & \quad = \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f_{1,j}(T_{1}^{P_{1}(n)}x)\cdots f_{k,j}(T_{k}^{P_{k}(n)}x)h_{i}(R^{\Omega(n)}\pi(x)) \nonumber\\ & \quad= \int_{Y}h_id\nu\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f_{1,j}(T_{1}^{P_{1}(n)}x)\cdots f_{k,j}(T_{k}^{P_{k}(n)}x) \nonumber\\ & \quad= \int_{Y}h_i\circ \pi d\mu_{y}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f_{1,j}(T_{1}^{P_{1}(n)}x)\cdots f_{k,j}(T_{k}^{P_{k}(n)}x). \end{align} $$
Based on (5.8), by a standard approximation argument, we know that (5.7) exists in
$L^{2}(\mu _{y})$
. This finishes the whole proof.
The proof of Theorem 1.10 is similar that of Theorem 1.9. The only difference between them is that we should use Theorem 5.1 in the proof of Theorem 1.10 instead of Theorem 5.2.
Let
$X\to \mathcal {M}(X),x\mapsto \mu _x$
be the ergodic decomposition of
$\mu $
with respect to S. Once one finishes the proof of Theorem 1.10, the following equality will appear:
$$ \begin{align} & \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}g_{1}(T^{P_{1}(n)}x)\cdots g_{k}(T^{P_{k}(n)}x)g_{k+1}(S^{\Omega(n)}x) \nonumber\\ & \quad = \int_{X}g_{k+1}d\mu_{x}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}g_{1}(T^{P_{1}(n)}x)\cdots g_{k}(T^{P_{k}(n)}x) \end{align} $$
in
$L^{2}(\mu )$
, where
$g_{1},\ldots ,g_{k+1}\in L^{\infty }(\mu )$
.
6 Proofs of Propositions 1.11–1.13
First, let us recall Furstenberg’s correspondence principle.
Theorem 6.1. (See [Reference Furstenberg12, Theorem 1.1], [Reference Frantzikinakis, Host and Kra10, §2.1])
Let
$k\in \mathbb {N}$
and
$E\subset \mathbb Z^{k}$
. There exists a Lebesgue space
$(X,\mathcal {X},\mu )$
, commuting invertible measure-preserving transformations
$T_1,\ldots ,T_k:X\to X$
, and
$A\in \mathcal {X}$
with
$\mu (A)=d^{*}(E)$
such that
$$ \begin{align*}d^{*}(E\cap (E-d_1)\cap \cdots \cap (E-d_m))\ge \mu\bigg(A\cap \prod_{i=1}^{k}T_{i}^{-d_{1,i}}A\cdots \cap \prod_{i=1}^{k}T_{i}^{-d_{m,i}}A\bigg)\end{align*} $$
for all
$m\in \mathbb {N}$
and all
$d_1=(d_{1,1},\ldots ,d_{1,k}),\ldots ,d_m=(d_{m,1},\ldots ,d_{m,k})\in \mathbb Z^{k}$
.
Based on Theorem 6.1, Proposition 1.11 can be deduced from the following result.
Proposition 6.2. Let
$P_1,\ldots ,P_k$
be pairwise independent polynomials with integer coefficients and zero constant terms, and any one of them is not of the form
$an^d$
. Let
$S,T_1,\ldots ,T_k$
be a family of invertible measure-preserving transformations acting on the Lebesgue space
$(X,\mathcal {X},\mu )$
and
$T_1,\ldots ,T_k$
are commuting. Then, for every
$A\in \mathcal {X}$
with
$\mu (A)>0$
, there exists a positive constant c, depending only on
$\mu (A)$
and
$P_1,\ldots ,P_k$
, such that
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\mu(A\cap T_{1}^{-P_{1}(n)}A\cap \cdots\cap T_{k}^{-P_{k}(n)}A\cap S^{-\Omega(n)}A)\ge c.\end{align*} $$
Likely, Propositions 1.12 and 1.13 can be deduced from a similar result.
Proposition 6.3. Let
$P_{1},\ldots ,P_{k}\in \mathbb Z[n]$
with zero constant terms. Let
$S,T$
be two invertible measure-preserving transformations acting on the Lebesgue space
$(X,\mathcal {X},\mu )$
. Then, for every
$A\in \mathcal {X}$
with
$\mu (A)>0$
, there exists a positive constant c, depending only on
$\mu (A)$
and
$P_1,\ldots ,P_k$
, such that
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\mu(A\cap T^{-P_{1}(n)}A\cap \cdots\cap T^{-P_{k}(n)}A\cap S^{-\Omega(n)}A)\ge c.\end{align*} $$
Before proving the above propositions, we introduce a quantitative version of the polynomial Szemerédi theorem.
Theorem 6.4. [Reference Frantzikinakis, Host and Kra10, Theorem 4.1]
Let
$P_{1},\ldots ,P_{k}\in \mathbb Z[n]$
with zero constant terms. Let
$T_1,\ldots ,T_k$
be a family of commuting invertible measure-preserving transformations acting on the Lebesgue space
$(X,\mathcal {X},\mu )$
. Then, for every
$A\in \mathcal {X}$
with
$\mu (A)>0$
, there exists a positive constant c, depending only on
$\mu (A)$
and
$P_1,\ldots ,P_k$
, such that
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\mu(A\cap T_{1}^{-P_{1}(n)}A\cap \cdots\cap T_{k}^{-P_{k}(n)}A)\ge c.\end{align*} $$
Now, we are about to prove Proposition 6.2.
Proof of Proposition 6.2
Fix
$\delta \in (0,1)$
. Fix a Lebesgue space
$(X,\mathcal {X},\mu )$
, a family of commuting invertible measure-preserving transformations
$T_1,\ldots ,T_k$
acting on it and
$A\in \mathcal {X}$
with
$\mu (A)=\delta $
. Fix an invertible measure-preserving transformation S acting on
$(X,\mathcal {X},\mu )$
. By Theorem 6.4, there exists a constant
$c(\delta )\in (0,1)$
, depending only on
$\delta $
and
$P_1,\ldots ,P_k$
, such that
$$ \begin{align} \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\mu(A\cap T_{1}^{-P_{1}(n)}A\cap \cdots\cap T_{k}^{-P_{k}(n)}A)\ge c(\delta). \end{align} $$
Then, by (6.1) and Theorem 5.2,
$$ \begin{align} \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\int_{X}1_{A}(x)\cdot \prod_{j=1}^{k}\mathbb E_{\mu}(1_A|\mathcal{Z}_{\infty}(T_j))(T_{j}^{P_{j}(n)}x)\,d\mu(x)\ge c(\delta). \end{align} $$
By Theorem 2.2 and [Reference Host and Kra16, Theorem 14.15], for each
$1\le i\le k$
, there exists a sequence of functions
$\{\phi _{i,j}\}_{j\ge 1}$
such that the following hold:
-
(1) for any
$1\le i\le k,j\ge 1$
and
$\mu $
-a.e.
$x\in X$
,
$\{\phi _{i,j}(T_{i}^{P_{i}(n)}x)\}_{n\in \mathbb Z}$
is a nilsequence; -
(2) for each
$1\le i\le k,j\ge 1$
,
$\Vert \mathbb E_{\mu }(1_{A}|\mathcal {Z}_{\infty }(T_i))-\phi _{i,j}\Vert _{L^{2k}(\mu )}\le 1/16^{jk}$
; -
(3) for each
$1\le i\le k,j\ge 1$
,
$0\le \phi _{i,j}\le 1$
.
Then, there exists a sufficiently large
$j_0$
such that
$$ \begin{align} \lim_{N\to\infty}\bigg\Vert{\frac{1}{N}\sum_{n=1}^{N}\bigg(\prod_{i=1}^{k}\mathbb E_{\mu}(1_A|\mathcal{Z}_{\infty}(T_i))(T_{i}^{P_{i}(n)}x)- \prod_{i=1}^{k}\phi_{i,j_0}(T_{i}^{P_{i}(n)}x)\bigg)}\bigg\Vert_{L^{2}(\mu)}< c(\delta)^{3}/32. \end{align} $$
Let
$X\to \mathcal {M}(X),y\mapsto \mu _y$
be the ergodic decomposition of
$\mu $
with respect to S. Note that the product of finitely many nilsequences is still a nilsequence. Then, we can define
$G:X\to [0,1]$
by putting
$$ \begin{align*}G(y)=\int_{X}1_{A}(x)\cdot \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\prod_{i=1}^{k}\phi_{i,j_0}(T_{i}^{P_{i}(n)}x)\,d\mu_{y}(x) \end{align*} $$
for
$\mu $
-a.e.
$y\in X$
. Clearly,
$$ \begin{align*}\int_{X}G(y)\,d\mu(y)=\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\int_{X}1_{A}(x)\cdot\prod_{i=1}^{k}\phi_{i,j_0}(T_{i}^{P_{i}(n)}x)\,d\mu(x).\end{align*} $$
$$ \begin{align*}\int_{X}G(y)\,d\mu(y)\ge 7c(\delta)/8.\end{align*} $$
So,
$$ \begin{align} 7c(\delta)/8&\le \int_{X}G(y)\,d\mu(y) \notag \\ & = \int_{\{y\in X:G(y)>c(\delta)/2\}}G(y)\,d\mu(y)+\int_{\{y\in X:G(y)\le c(\delta)/2\}}G(y)\,d\mu(y) \notag \\ & \le \mu(\{y\in X:G(y)>c(\delta)/2\})+c(\delta)/2. \end{align} $$
By (6.4),
$$ \begin{align} \mu(E:=\{y\in X:G(y)>c(\delta)/2\})\ge 3c(\delta)/8. \end{align} $$
By Theorem 1.9, we know that the limit
$$ \begin{align*}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\mu(A\cap T_{1}^{-P_{1}(n)}A\cap \cdots\cap T_{k}^{-P_{k}(n)}A\cap S^{-\Omega(n)}A)\end{align*} $$
exists.
Next, we begin to estimate the uniform lower bound,
$$ \begin{align*} & \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\mu(A\cap T_{1}^{-P_{1}(n)}A\cap \cdots\cap T_{k}^{-P_{k}(n)}A\cap S^{-\Omega(n)}A) \\&= \lim_{N\to\infty}\int_{X}1_{A}(x)\cdot\bigg( \frac{1}{N}\sum_{n=1}^{N}1_{A}(T_{1}^{P_{1}(n)}x)\cdots 1_{A}(T_{k}^{P_{k}(n)}x)\cdot 1_{A}(S^{\Omega(n)}x)\bigg)\,d\mu(x) \\&= \lim_{N\to\infty}\int_{X}1_{A}(x)\cdot \mu_{x}(A)\cdot\bigg( \frac{1}{N}\sum_{n=1}^{N}\prod_{j=1}^{k}1_{A}(T_{j}^{P_{j}(n)}x)\bigg)\,d\mu(x)\ (({5.1})) \\&= \lim_{N\to\infty}\int_{X}\!1_{A}(x)\cdot \mu_{x}(A)\cdot\bigg(\! \frac{1}{N}\!\sum_{n=1}^{N}\prod_{j=1}^{k}\!\mathbb E_{\mu}(1_A|\mathcal{Z}_{\infty}(T_j))(T_{j}^{P_{j}(n)}x)\!\bigg)\,d\mu(x)\ (\text{Theorem }{5.2}) \\&\ge \lim_{N\to\infty}\bigg|\frac{1}{N}\!\sum_{n=1}^{N}\int_{X}\!1_{A}(x)\cdot \mu_{x}(A)\cdot \prod_{i=1}^{k}\phi_{i,j_0}(T_{i}^{P_{i}(n)}x)\,d\mu(x)\bigg| - \lim_{N\to\infty}\bigg|\!\int_{X}\!1_{A}(x)\cdot \mu_{x}(A) \\&\quad \times \frac{1}{N}\sum_{n=1}^{N}\bigg(\prod_{i=1}^{k}\mathbb E_{\mu}(1_A|\mathcal{Z}_{\infty}(T_i))(T_{i}^{P_{i}(n)}x)- \prod_{i=1}^{k}\phi_{i,j_0}(T_{i}^{P_{i}(n)}x)\bigg)\,d\mu(x)\bigg| \\&\ge \lim_{N\to\infty}\bigg|\frac{1}{N}\sum_{n=1}^{N}\int_{X}1_{A}(x)\cdot \mu_{x}(A)\cdot \prod_{i=1}^{k}\phi_{i,j_0}(T_{i}^{P_{i}(n)}x)\,d\mu(x)\bigg| - c(\delta)^{3}/32\ (({6.3})) \\&= \int_{X}\mu_{y}(A) \int_{X}1_{A}(x)\cdot \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\prod_{i=1}^{k}\phi_{i,j_0}(T_{i}^{P_{i}(n)}x)\,d\mu_{y}(x)\,d\mu(y)- c(\delta)^{3}/32 \\&= \int_{X}\mu_{y}(A)G(y)\,d\mu(y)- c(\delta)^{3}/32 \\&\ge \int_{E}G(y)^{2}d\mu(y)- c(\delta)^{3}/32\ (\text{by the fact that}\ G(y)\le \mu_{y}(A)) \\&\ge \frac{1}{16}c(\delta)^{3}.\ (({6.5})) \end{align*} $$
This finishes the whole proof.
The proof of Proposition 6.3 is similar to that of Proposition 6.2. The only difference is that we should use Theorems 1.10, 5.1 and (5.9) in the proof of Proposition 6.3 instead of Theorems 1.9, 5.2 and (5.1).
7 Some questions
7.1 On Proposition 1.11
Due to those restrictions for polynomials in Theorem 1.9, we cannot give an answer to the following question.
Question 7.1. Fix
$k\ge 2$
and
$A\subset \mathbb {N}^{k+1}$
with positive upper Banach density. Are there
$a\in \mathbb {N}^{k+1},d\in \mathbb {N}$
such that
$$ \begin{align*}a,a+d\vec{e}_{1},\ldots,a+kd\vec{e}_{k},a+\Omega(d)\vec{e}_{k+1}\in A\ ?\end{align*} $$
So, we ask a special case of the above question here.
Question 7.2. Fix
$k\ge 2$
. Is it true that for any finite colouring of
$\mathbb {N}^{k+1}$
, there are
${a\in \mathbb {N}^{k+1},d\in \mathbb {N}}$
such that the set
$$ \begin{align*}\{a,a+d\vec{e}_{1},\ldots,a+kd\vec{e}_{k},a+\Omega(d)\vec{e}_{k+1}\}\end{align*} $$
is monochromatic?
7.2 On recurrence times
As a direct result of Theorem 6.1 and Proposition 6.3, we know that for any
$\delta>0$
, there is
$c(\delta )>0$
, depending only on
$\delta $
, such that for any
$E\subset \mathbb {N}$
with
$d^{*}(E)=\delta $
, then
$$ \begin{align} \liminf_{N\to \infty}\frac{1}{N}\sum_{n=1}^{N}d^{*}(E\cap (E-\Omega(n)))\ge c(\delta). \end{align} $$
In [Reference Frantzikinakis, Host and Kra9], by combining some number theory results and a quantitative version of the Roth theorem, Frantzikinakis, Host and Kra proved that for any
$E\subset \mathbb {N}$
with positive upper Banach density, there are infinitely many n in
$\mathbb {P}-1(\mathbb {P}+1)$
such that
$$ \begin{align*}d^{*}(E\cap (E-n)\cap (E-2n))>0.\end{align*} $$
Later, Wooley and Ziegler [Reference Wooley and Ziegler21] extended it to general polynomials with zero constant terms. Based on these results and (7.1), we expect a similar result for
$\Omega (n)$
here.
Question 7.3. Fix
$E\subset \mathbb {N}$
with positive upper Banach density. Are there infinitely many n in
$\mathbb {P}-1(\mathbb {P}+1)$
such that
$$ \begin{align*}\Omega(n)\in (E-E)?\end{align*} $$
If Question 7.3 has a positive answer, then by letting E be the arithmetic progressions with infinite length, we have that for each
$k\in \mathbb {N}$
, there are infinitely many n in
${\mathbb {P}-1(\mathbb {P}+1)}$
such that
$k|\Omega (n)$
.
Acknowledgements
R.X. is supported by National Natural Science Foundation of China (123B2007,12371196). The initial ideas of the paper first arose in an online seminar held during the winter of 2023. The author’s thanks go to Zhengxing Lian for organizing this seminar and the referee for useful remarks and suggestions.
