1 Introduction
Our goal is to resolve a conjecture posed in [Reference Bergelson, Moreira and Richter1, Remark 1.12] by showing that multiple ergodic averages of the form
converge to a positive limit as
$N\to \infty $
whenever f is a tempered function; that is, if there exists
$\ell \in \mathbb {N}$
such that
$f^{(\ell )}$
decreases to
$0$
and
$\lim _{x\to \infty }xf^{(\ell )}(x)=\infty $
. We will prove this by developing some new tools regarding weighted averages and applying a theorem from [Reference Bergelson, Moreira and Richter1].
In order to discuss this in more detail, we first require some definitions. Let
$\Delta $
denote the discrete derivative, which acts on a function f defined on
$\mathbb {N}=\{1,2,\ldots \}$
by
$\Delta f(n)= f(n)-f(n-1)$
for
$n\geqslant 2$
and
$\Delta f(1) = f(1)$
. For
$\ell \geqslant 1$
, we define
$\Delta ^{\ell }f(n) = \Delta (\Delta ^{\ell -1}f(n))$
, where
$\Delta ^0f(n) = f(n)$
. For
$p(x) = a_mx^m+\cdots +a_1x+a_0\in \mathbb {Z}[x]$
, we put
Definition 1.1. Let
$\mathscr {F}_1$
be the collection of functions defined by
Having defined
$\mathscr {F}_{\ell }$
for some
$\ell \in \mathbb {N}$
, let
$\mathscr {F}_{\ell +1} = \{f:\mathbb {N}\rightarrow \mathbb {R}:\Delta f \in \mathscr {F}_{\ell }\}$
. Finally, let
$\mathscr {F} =~\bigcup _{\ell =1}^{\infty }\mathscr {F}_{\ell }$
. We may note that
$\mathscr {F}$
contains all tempered functions.
Remark 1.1. A function
$f:\mathbb {N}\rightarrow \mathbb {R}$
is contained in
$\mathscr {F}$
if and only if there is an
$\ell \in \mathbb {N}$
such that
$\Delta ^{\ell } f$
is eventually decreasing to
$0$
but
$\Delta ^{\ell -1}f$
tends to
$\infty $
. So any function of the form
$f(n) = n^c$
for
$c\in (0,\infty )\backslash \mathbb {Z}$
is contained in
$\mathscr {F}$
, but no polynomial is contained in
$\mathscr {F}$
.
The following is Theorem D in [Reference Bergelson, Moreira and Richter1].
Theorem 1.2. [Reference Bergelson, Moreira and Richter1, Theorem D]
Let
$k \in \mathbb {N}, \ell \in \mathbb {N}$
,
$f \in \mathscr {F}_{\ell }$
. Let
$W= \Delta ^{\ell -1}f$
so that
$\lim _{n\to \infty }W(n)=\infty $
and
$\lim _{n\to \infty }\Delta W(n)=0$
. Let
$(X,\mathscr {B},\mu ,T)$
be an invertible probability-measure-preserving system and let
$p_1,\ldots ,p_k \in \mathbb {Z}[x]$
.
-
(a) For any $h_1,\ldots ,h_k \in L^{\infty }(X,\mathscr {B},\mu )$
, the limit $$ \begin{align*} \lim_{W(N)-W(M)\to\infty}\frac{1}{W(N)-W(M)}\sum_{n=M}^N\Delta W(n) T^{\lfloor p_1(\Delta)f(n) \rfloor}h_1\cdots T^{\lfloor p_k(\Delta)f(n) \rfloor}h_k \end{align*} $$exists in $L^2(X,\mathscr {B},\mu )$
.
-
(b) For any $A \in \mathscr {B}$
with
$\mu (A)> 0$
, the limit $$ \begin{align*} \lim_{W(N)-W(M)\to\infty}\frac{1}{W(N)-W(M)}\sum_{n=M}^N\Delta W(n)\mu\bigg(A\cap\bigg(\bigcap_{i=1}^{k}T^{-\lfloor p_i(\Delta)f(n) \rfloor}A\bigg)\bigg) \end{align*} $$exists and is positive.
In [Reference Bergelson, Moreira and Richter1, Remark 1.12], it is conjectured that each limit of weighted averages appearing in Theorem 1.2 can be replaced with the corresponding limit of Cesàro averages whenever f is a tempered function. Theorem 1.2 applies when f is any element of
$\mathscr {F}$
, not just when f is a tempered function. However, when f is not tempered the weighted averages in Theorem 1.2 cannot be replaced by Cesàro averages (cf. Example 5.2 where
$f(n)=\log (n)$
is not tempered).
The following theorem is a corollary of our main result and it gives an affirmative answer to the conjecture in [Reference Bergelson, Moreira and Richter1].
Theorem A. Let
$f\in \mathscr {F}_{\ell }$
be a tempered function. Let
$W=\Delta ^{\ell -1}f$
so that W increases to
$\infty $
,
$\Delta W$
decreases to
$0$
, and
$\lim _{N\to \infty }N\cdot \Delta W(N) = \infty $
. Suppose that
$(x_n)_{n\in \mathbb {N}}$
is a bounded sequence in a Banach space Y and let
$L\in Y$
. If
then
Thus, in light of Theorem A, the following theorem follows immediately from Theorem 1.2.
Theorem 1.3. Let
$(X,\mathscr {B},\mu ,T)$
be an invertible probability-measure-preserving system. Let
$f\in \mathscr {F}$
be a tempered function, let
$k\in \mathbb {N}$
, and let
$p_1,\ldots ,p_k \in \mathbb {Z}[x]$
.
-
(a) For any $h_1,\ldots ,h_k \in L^{\infty }(X,\mathscr {B},\mu )$
, the limit (1.2) $$ \begin{align} \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^NT^{\lfloor p_1(\Delta)f(n) \rfloor}h_1\cdots T^{\lfloor p_k(\Delta)f(n) \rfloor}h_k \end{align} $$exists in $L^2(X,\mathscr {B},\mu )$
.
-
(b) For any $A \in \mathscr {B}$
with
$\mu (A)> 0$
, the limit (1.3) $$ \begin{align} \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N\mu(A\cap T^{-\lfloor p_1(\Delta)f(n) \rfloor}A\cap\cdots \cap T^{-\lfloor p_k(\Delta)f(n) \rfloor}A) \end{align} $$exists and is positive.
Using the identity
$f(n-k) = (1-\Delta )^kf(n)$
, we obtain a special case of Theorem 1.3, which not only extends Theorem 1.2 but also is an extension of [Reference Frantzikinakis5, Theorem 5.6].
Corollary 1.4. Let
$(X,\mathscr {B},\mu ,T)$
be an invertible probability-measure-preserving system. Let
$f\in \mathscr {F}$
be a tempered function and let
$k\in \mathbb {N}$
.
-
(a) For any $h_1,\ldots ,h_k \in L^{\infty }(X,\mathscr {B},\mu )$
, the limit (1.4) $$ \begin{align} \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^NT^{\lfloor f(n) \rfloor}h_1\cdots T^{\lfloor f(n+k) \rfloor}h_k \end{align} $$exists in $L^2(X,\mathscr {B},\mu )$
.
-
(b) For any $A \in \mathscr {B}$
with
$\mu (A)> 0$
, the limit (1.5) $$ \begin{align} \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N\mu(A\cap T^{-\lfloor f(n) \rfloor}A\cap\cdots\cap T^{-\lfloor f(n+k) \rfloor}A) \end{align} $$exists and is positive.
We will now introduce some notation in order to formulate our main theorem.
Definition 1.2. Let
$W:\mathbb {N}\rightarrow \mathbb {R}$
be a function which increases to
$\infty $
and let
$(x_n)_{n\in \mathbb {N}}$
be a bounded sequence in a Banach space Y.
-
• For $N\in \mathbb {N}$
, define the Nth W-weighted average of
$(x_n)_{n\in \mathbb {N}}$
by (1.6) $$ \begin{align} {\mathbb{E}}_{n\leqslant N}^Wx_n= \frac{1}{W(N)}\sum_{n=1}^N\Delta W(n) x_n. \end{align} $$
-
• More generally, we define the W-weighted average of $(x_n)_{n\in \mathbb {N}}$
over the interval
$[M,N]$
to be (1.7) $$ \begin{align} {\mathbb{E}}_{n\in [M,N]}^Wx_n = \frac{1}{W(N)-W(M)}\sum_{n=M}^N\Delta W(n) x_n. \end{align} $$
-
• Define the uniform W-weighted average of $(x_n)_{n\in \mathbb {N}}$
as (1.8) $$ \begin{align} {\mathbb{E}}_{\mathrm{unif}}^W(x_n) =\lim_{W(N)-W(M)\to \infty}{\mathbb{E}}_{n\in [M,N] }^W(x_n) \end{align} $$when the limit exists. The limit $\lim _{W(N)-W(M)\to \infty }$
is taken over all sequences of intervals
$[M_j,N_j]$
for
$j\in \mathbb {N}$
, with
$W(N_j)-W(M_j)\to \infty $
as
$j\to \infty $
.
-
• Define the iterated W-weighted averages of $(x_n)_{n\in \mathbb {N}}$
by
$ {\mathbb {E}}(1)_{n\leqslant N}^Wx_n = {\mathbb {E}}_{n\leqslant N}^Wx_n$
and (1.9) $$ \begin{align} {\mathbb{E}}(k+1)_{n\leqslant N}^Wx_n = {\mathbb{E}}_{n\leqslant N}^W({\mathbb{E}}(k)_{m\leqslant n}^Wx_m) = \frac{1}{W(N)}\sum_{n=1}^N\Delta W(n) \cdot{\mathbb{E}}(k)_{m\leqslant n}^Wx_m \end{align} $$for $k\in \mathbb {N}$
.
-
• For $L\in Y$
, we say that
$ {\mathbb {E}}(\infty )^Wx_n = L$
if (1.10) $$ \begin{align} \lim_{k\to\infty}\limsup_{N\to\infty}|{\mathbb{E}}(k)_{n\leqslant N}^W(x_n -L)|=0. \end{align} $$
Example 1.5. When
$W(N) = N$
,
$ {\mathbb {E}}_{n\leqslant N}^Wx_n$
denotes the usual Cesàro averages and we will write
$ {\mathbb {E}}_{n\in [M,N]}x_n$
instead of
$ {\mathbb {E}}^W_{n\in [M,N]}x_n$
so that we have
$ {\mathbb {E}}_{n\in [M,N]}x_n = {\mathbb {E}}_{n\in [M,N]}^Wx_n = ({1}/({N-M}))\sum _{n=M}^{N}x_n$
. When
$U(N) = \log (N)$
,
$ {\mathbb {E}}_{n\leqslant N}^Ux_n$
is the logarithmic average
It will be useful to consider weighted averages with respect to functions which grow faster than any polynomial. For example, let
$V(x) = e^{\sqrt {x}}$
. Then
Example 1.6. When
$W(N) = N$
, the uniform W-weighted average of
$(x_n)_{n\in \mathbb {N}}$
is the usual notion of the uniform Cesàro limit
and
$ {\mathbb {E}}_{ \mathrm{unif}}x_n = L$
is equivalent to the statement that
$\lim _{N\to \infty }({1}/{N})\sum _{n=k}^{N+k}x_n$
converges to L uniformly in k.
When
$U(N) = \log (N)$
, the uniform U-weighted average of
$(x_n)_{n\in \mathbb {N}}$
is
and
$ {\mathbb {E}}_{ \mathrm{unif}}^{\log }x_n =L$
is equivalent to the statement that
$\lim _{a\to \infty }({1}/{\log a}) \sum _{n=k}^{ak}({x_n}/{n})$
converges to L uniformly in k.
We now show that Theorem A follows from our main result, Theorem B.
Theorem B. Let
$V:\mathbb {N}\rightarrow (0,\infty )$
be a function which increases to
$\infty $
with
$\Delta {\log} (V(N))$
eventually decreasing to
$0$
such that
$\lim _{N\to \infty }({\Delta {\log} (V(N)})/{\Delta {\log} (N)})=\infty $
. Let Y be a Banach space, let
$(x_n)_{n\in \mathbb {N}}\subseteq Y$
be a bounded sequence, and let
$L\in Y$
. Consider the following statements:
-
(1) $\lim _{N\to \infty } {\mathbb {E}}_{n\leqslant N}^Vx_n =L$
; -
(2) $ {\mathbb {E}}(\infty )^Vx_n = L$
; -
(3) $\lim _{N\to \infty } {\mathbb {E}}_{n\leqslant N}^Ux_n = L$
for each function
$U:\mathbb {N}\rightarrow (0,\infty )$
which increases to
$\infty $
and satisfies (1.15) $$ \begin{align} \lim_{N\to\infty}\frac{V(N-1)}{\Delta U(N)}\bigg(\frac{\Delta U(N)}{\Delta V(N)}-\frac{\Delta U(N-1)}{\Delta V(N-1)}\bigg)=-1; \end{align} $$
-
(4) $\lim _{N\to \infty } {\mathbb {E}}_{n\in [N-s(N),N]}x_n = L$
for each non-decreasing function
$s:\mathbb {N}\rightarrow \mathbb {N}$
satisfying (1.16) $$ \begin{align} \lim_{N\to\infty}s(N) \cdot \Delta{\log}(V(N))=\infty \end{align} $$and $s(N)\leqslant N-1$
for all
$N\in \mathbb {N}$
;
-
(5) $ {\mathbb {E}}_{\mathrm {unif}}^{\log V}x_n = L$
.
We have that
Proof of Theorem A.
Let
$V(N) = e^{W(N)}$
and suppose that
that is,
$ {\mathbb {E}}_{ \mathrm{unif}}^Wx_n = L$
and so condition (5) of Theorem B holds. We have that
$\Delta{\log} (V(N))$
is decreasing to
$0$
and that
because of our assumptions on W. Thus we can apply Theorem B to deduce that condition (4) holds.
Let
$s(N) = N-1$
so that
$ {\mathbb {E}}_{n\in [N-s(N),N]}x_n = {\mathbb {E}}_{n\in [1,N]}x_n = {\mathbb {E}}_{n\leqslant N}x_n$
for all
$N\in \mathbb {N}$
and
$\lim _{N\to \infty }s(N)\cdot \Delta{\log}(V(N)) = \lim _{N\to \infty }N\cdot \Delta W(N)=\infty $
. By Theorem B, we have that
$ {\mathbb {E}}_{n\in [N-s(N),N]}x_n=L$
. This completes the proof.
Remark 1.7. The limit in equation (1.15) initially appears unwieldy and impractical. However, let us assume that the limit in equation (1.15) exists. Then we can perform some algebraic manipulations and apply the Stolz–Cesàro theorem [Reference Pólya and Szegő7, Problem 70] to see that
So the limit in equation (1.15) is equal to
$-1$
if and only if
$\lim _{N\to \infty }(({V(N)\cdot \Delta U(N)})/ ({U(N)\cdot \Delta V(N)}))=0$
. We can obtain a clearer criterion when we use our assumption that
$\lim _{N\to \infty }\Delta{\log}(V(N))=0$
, since we have
and so
$\lim _{N\to \infty }\Delta{\log}\ V(N) = 0$
if and only if
$\lim _{N\to \infty }({\Delta V(N)}/{V(N)})=0$
. It follows that
One more application of the Stolz–Cesàro theorem shows that if
$\lim _{N\to \infty }({\Delta{\log}(U(N))}/ {\Delta{\log}(V(N))})=0$
then
$\lim _{N\to \infty }({ \log (U(N))}/{ \log (V(N))})=0$
. In particular, when the limit in equation (1.15) is known to exist, the assumption that equation (1.15) holds can be replaced with the assumption that
$\lim _{N\to \infty }({\log (U(N))}/{\log (V(N))})=0$
. For one important example of this, recall that a Hardy field is a field of real-valued functions which is closed under derivation, with the equivalence relation that two functions are equal if they are equal outside of a compact set. What is important for our purposes is any function contained in a Hardy field is eventually monotone and so any limit involving functions contained in the same Hardy field will exist in
$\mathbb {R}\cup \{-\infty ,+\infty \}$
(for more details, see [Reference Boshernitzan3]). When U and V are contained in the same Hardy field, the limit in (1.15) must exist.
Remark 1.7 gives a special case of Theorem B.
Theorem C. Let V be a function which is contained in a Hardy field and tends to
$\infty $
. Suppose that
$\lim _{N\to \infty }({ \log (V(N))}/{ \log (N)})=\infty $
and
$\lim _{N\to \infty }({\log (V(N))}/{ N})=0$
. Let Y be a Banach space, let
$(x_n)_{n\in \mathbb {N}}\subseteq Y$
be a bounded sequence, and let
$L\in Y$
. Consider the following statements:
-
(1) $\lim _{N\to \infty } {\mathbb {E}}_{n\leqslant N}^Vx_n =L$
; -
(2) $ {\mathbb {E}}(\infty )^Vx_n = L$
; -
(3) $\lim _{N\to \infty } {\mathbb {E}}_{n\leqslant N}^Ux_n = L$
for each function U which tends to
$\infty $
, satisfies
$\lim _{N\to \infty }(\log (U(N))/{\log (V(N))})=0 $
, and is contained in the same Hardy field as V; -
(4) $\lim _{N\to \infty } {\mathbb {E}}_{n\in [N-s(N),N]}x_n = L$
for each non-decreasing function
$s:\mathbb {N}\rightarrow \mathbb {N}$
satisfying
$ \lim _{N\to \infty }s(N) \cdot \Delta{\log}(V(N))=\infty $
and
$s(N)\leqslant N-1$
for all
$N\in \mathbb {N}$
; -
(5) $ {\mathbb {E}}_{\mathrm {unif}}^{\log V}x_n = L$
.
We have that
Of special interest in Theorem B is condition (4), since this allows us to relate weighted averages to Cesàro averages along a sequence of intervals.
Example 1.8. Let
$E\subseteq \mathbb {N}$
be a set with
Let
$f\in \mathscr {F}_{\ell }$
and let
$k\in \mathbb {N}$
. Let
$W= \Delta ^{\ell -1}f$
. For each
$n\in \mathbb {N}$
, let
$A(n)$
be the set of
$a\in \mathbb {N}$
such that
From [Reference Bergelson, Moreira and Richter1, Corollary F] we know that the set
is W-syndetic, meaning that
Using Theorem B, we may rephrase this W-syndetic condition as saying that
for any non-decreasing function
$s\colon \mathbb {N}\rightarrow \mathbb {N}$
with
$\lim _{N\to \infty }s(N)\cdot \Delta W(N)=\infty $
and
$s(N)\leqslant N-1$
for all
$N\in \mathbb {N}$
. More concretely, for the sake of example we can take
$f(n) = n^{3/2}$
. Fix
$k\in \mathbb {N}$
and let
$\varepsilon>0$
. We can take
$s(N) = N^{1/2+\varepsilon }$
in equation (1.19) to see that for all large enough N, there is an
$n\in [N-N^{1/2+\varepsilon },N]$
such that E contains a configuration of the form
and, moreover, these configurations are abundant in the sense of equation (1.18).
1.1 Structure of the paper
The remainder of the paper consists of a proof of Theorem B. It is clear that
$(1)\implies (2)$
by definition. We will prove the rest of the implications one by one. In §2 we prove
$(2)\implies (3)$
, in §3 we prove
$(3)\implies (4)$
, and in §4 we prove
$(4)\implies (5)$
. Finally, in §5 we prove that
$(5)\implies (2)$
.
2
$(2)\implies (3)$
In this section we prove the implication
$(2)\implies (3)$
in Theorem B. This implication will follow readily from the following lemma, whose proof appears in an unpublished manuscript authored by Michael Boshernitzan [Reference Boshernitzan4].
Lemma 2.1. Let
$V,U:\mathbb {N}\rightarrow (0,\infty )$
increase to
$\infty $
and suppose that
Let Y be a Banach space and let
$(x_n)_{n\in \mathbb {N}}\subseteq Y$
be a bounded sequence. Then
where the
$o_{N\to \infty }(1)$
term depends only on
$U,V$
and
$\sup _{n\in \mathbb {N}}\|x_n\|$
.
Proof. Recall that the summation by parts formula says that
for functions
$f,g$
on
$\mathbb {N}$
. By the hypothesis of the lemma, we may apply the results of Remark 1.7 to obtain
$\lim _{N\to \infty }(({V(N)\cdot \Delta U(N)})/({U(N)\cdot \Delta V(N)}))=0$
and hence
$\lim _{N\to \infty }({\Delta U(N)}/{U(N)})=0$
.
We will take the right-hand side of equation (2.1) and rewrite it in the form of the left-hand side. Put
$F(n) = \sum _{k=1}^n\Delta V(k)\cdot x_k$
so that
$\Delta F(n) = \Delta V(n)\cdot x_n$
and
${F(N)}/{V(N)}= {\mathbb {E}}_{n\leqslant N}^Vx_n$
.
Apply summation by parts to obtain
The first term in (2.5) is equal to
$(({V(N)\hspace{-1pt}\cdot\hspace{-1pt} \Delta U(N)})/({U(N)\hspace{-1pt}\cdot\hspace{-1pt} \Delta V(N)}))\hspace{-1pt}\cdot\hspace{-1pt} ({F(N)}/{V(N)})$
, which tends to
$0$
as
$N\to \infty $
since
${F(N)}/{V(N)} = {\mathbb {E}}_{n\leqslant N}^Vx_n$
is bounded in norm by
$\sup _{n\leqslant N}\|x_n\|$
. The second term in (2.5) is
By assumption, we have
$({V(n-1)}/{\Delta U(n)})({\Delta U(n)}/{\Delta V(n)}-{\Delta U(n-1)}/{\Delta V(n-1)}) = -1+o_{N\to \infty }(1)$
.
Note that
${F(n-1)}/{V(n-1)} = {\mathbb {E}}_{k\leqslant n-1}^Vx_k$
is bounded in norm by
$\sup _{n\in \mathbb {N}}\|x_n\|$
. So (2.5) is
and the
$o_{N\to \infty }(1)$
term will depend only on
$U,V$
, and
$\sup _{n\leqslant N}\|x_n\|$
, and in particular not on
$(x_n)_{n\in \mathbb {N}}$
. Finally, recalling that
$\lim _{N\to \infty }({\Delta U(N)}/{U(N)})=0$
, we have
Corollary 2.2. Let Y be a Banach space, let
$(x_n)_{n\in \mathbb {N}}\subseteq Y$
be a bounded sequence, and let
$L\in Y$
. Suppose that
$V:\mathbb {N}\rightarrow (0,\infty )$
is a function which increases to
$\infty $
with
$\limsup _{N\to \infty }({\Delta V(N)}/{V(N)})<1$
. Let
$U:\mathbb {N}\rightarrow (0,\infty )$
be a function which increases to
$\infty $
and satisfies
If
$ {\mathbb {E}}(\infty )^Vx_n = L$
then
$\lim _{N\to \infty } {\mathbb {E}}_{n\leqslant N}^Ux_n = L$
.
Proof. Suppose that
$ {\mathbb {E}}(\infty )^Vx_n = L$
. Let
$\varepsilon>0$
. We can pick
$K\in \mathbb {N}$
such that
$| {\mathbb {E}}_{n\leqslant N}^V(k)(x_n)-L|<\varepsilon $
for all large enough N. Now consider
$ {\mathbb {E}}_{n\leqslant N}^U( {\mathbb {E}}_{k\leqslant n}^V(K)x_k)$
. From equation (2.1),
Using linearity, we have
But we can bound
$| {\mathbb {E}}_{n\leqslant N}^U( {\mathbb {E}}_{k\leqslant n}^V(K)x_k-L)|$
using the triangle inequality,
Taking
$\limsup _{N\to \infty }$
of equation (2.6), we find that
This holds for each
$\varepsilon>0$
, so we conclude that
$\lim _{N\to \infty } {\mathbb {E}}_{n\leqslant N}^Ux_n=L$
.
3
$(3)\implies (4)$
In this section we prove the implication
$(3)\implies (4)$
in Theorem B. We begin by recalling the Silverman–Toeplitz theorem [Reference Boos and Cass2, Theorem 2.3.7].
Theorem 3.1. (Silverman–Toeplitz)
Let
$(c_{N,n})_{N,n\in \mathbb {N}}$
be a doubly indexed sequence of complex numbers and let Y be a Banach space. The following assertions are equivalent:
-
(A) $\lim _{N\to \infty }\sum _{n\in \mathbb {N}}c_{N,n}y_n = \lim _{n\to \infty }y_n$
for each convergent sequence
$(y_n)_{n\in \mathbb {N}}\subseteq Y$
; -
(B) $(c_{N,n})_{N,n\in \mathbb {N}}$
satisfies each of the following:-
• for each $n\in \mathbb {N}$
,
$\lim _{N\to \infty }c_{N,n}=0$
, -
• $\lim _{N\to \infty }\sum _{n\in \mathbb {N}}c_{N,n}=1$
, -
• $\limsup _{N\to \infty }\sum _{n\in \mathbb {N}}|c_{N,n}|<\infty $
.
-
The Silverman–Toeplitz theorem will allow us to deduce the desired implication once we show that there is a sequence
$(c_{N,n})_{N,n\in \mathbb {N}}$
relating the averages of the form
$ {\mathbb {E}}_{n\in [N-s(N),N]}$
and the averages of the form
$ {\mathbb {E}}_{n\leqslant N}^U$
.
Lemma 3.2. Let Y be a Banach space, let
$U:\mathbb {N}\rightarrow (0,\infty )$
be an increasing function which satisfies
$\limsup _{N\to \infty }({\Delta{\log}(U(N))}/{\Delta{\log}(N)})=\infty $
and
$\limsup _{N\to \infty } (\Delta U(N)/{U(N)})=0$
. Additionally, assume that
$\Delta U$
is non-decreasing. Let
$s:\mathbb {N}\rightarrow \mathbb {N}$
be a non-decreasing function with
$s(N)\leqslant N-1$
for all
$N\in \mathbb {N}$
and
Then there exists a doubly indexed non-negative sequence
$(c_{N,n})_{N,n\in \mathbb {N}}$
which satisfies each of the conditions in item (B) of Theorem 3.1 such that for any bounded sequence
$(x_n)_{n\in \mathbb {N}}\subseteq Y$
,
Proof. Define
$(c_{N,n})_{N,n\in \mathbb {N}}$
by
and note that each
$c_{N,n}$
is non-negative since
$\Delta U$
is non-decreasing. Additionally, it is clear that
$\lim _{N\to \infty }c_{N,n}=0$
for each fixed n since
$s(N)\to \infty $
as
$N\to \infty $
. Next we will show that equation (3.1) holds. Switching the order of summation, we have
We will show the following two equations, from which equation (3.1) follows:
and
For (3.2), if
$k=N$
we have
and if
$N-s(N)\leqslant k<N$
then
Turning our attention to (3.3), we can observe that
$c_{N,k} = 0$
for
$k<N-s(N)$
and so as above we have
Then
Recall that
$ {\mathbb {E}}_{n\leqslant N-s(N)-1}^Ux_n$
is bounded and that
${U(N-s(N)-1)}/{U(N)}\leqslant 1$
since U is increasing. Additionally, from equation (1.17),
$ \lim _{N\to \infty }({\Delta{\log} U(N)}/({\Delta U(N)}/ {U(N)})) =1$
. By assumption, we have
$\lim _{N\to \infty }s(N)\cdot \Delta{\log}\ U(N)=\infty $
and so taking the limit of equation (3.5) shows that equation (3.3) holds. Lastly, we have
$\lim _{N\to \infty }\sum _{n\in \mathbb {N}}c_{N,n} = 1$
by taking
$x_n = 1$
for all
$n\in \mathbb {N}$
in equation (3.1). This concludes the proof.
Corollary 3.3. Let Y be a Banach space, let
$(x_n)_{n\in \mathbb {N}}\subseteq Y$
be a bounded sequence, and let
$L\in Y$
. Let
$V:\mathbb {N}\rightarrow (0,\infty )$
be an increasing function such that
$\Delta{\log}(V(N))$
decreases to
$0$
with
$\lim _{N\to \infty }({\Delta{\log}\ V(N)}/{\Delta{\log}(N)})=\infty $
. Suppose that
$\lim _{N\to \infty } {\mathbb {E}}_{n\leqslant N}^Ux_n = L$
for each function
$U:\mathbb {N}\rightarrow (0,\infty )$
which increases to
$\infty $
and satisfies
Let
$s:\mathbb {N}\rightarrow \mathbb {N}$
be a non-decreasing function satisfying
$\lim _{N\to \infty }s(N) \cdot \Delta{\log}(V(N))=~\infty $
and
$s(N)\leqslant N-1$
for all
$N\in \mathbb {N}$
. Then
$ {\mathbb {E}}_{n\in [N-s(N),N]}x_n = L$
.
Proof. Our strategy is to find functions U and
$\tilde {U}$
where U satisfies (1.15),
$\tilde {U}$
satisfies the assumptions of Theorem 3.2, and
Then by taking limits of equation (3.1) and invoking the Silverman–Toeplitz theorem it will follow that
$\lim _{N\to \infty } {\mathbb {E}}_{n\in [N-s(N),N]}x_n = L$
. We begin by picking a function
$r:\mathbb {N}\rightarrow (0,\infty )$
which decreases to
$0$
slowly enough such that:
-
(i) $\lim _{N\to \infty }r(N)\cdot \log (V(N))=\infty $
; -
(ii) $\lim _{N\to \infty }r(N)\cdot s(N)\cdot \Delta{\log}\ V(N) =\infty $
; -
(iii) $\lim _{N\to \infty }({\Delta{\log}(r(N))}/{\Delta{\log}\log (V(N))}){\kern-1pt}={\kern-1pt}\lim _{N\to \infty }(({\log (V(N)){\kern-1pt}\cdot{\kern-1pt} \Delta r(N)})/(r(N){\kern-1pt}\cdot \Delta{\log} (V(N))))=0$
, from which it follows that
$\lim _{N\to \infty }({\Delta{\log}(r(N))}/ \Delta{\log}(V(N)^{r(N)}))=0$
and
$\lim _{N\to \infty }({\log (V(N))\cdot \Delta r(N)}/{\Delta{\log}(V(N))})= \lim _{N\to \infty } ({\Delta{\log}(V(N)^{r(N)})}/{\Delta{\log}(V(N))})=0$
.
For example, we can take
Let
$U(N) = \sum _{n\leqslant N}(({\Delta V(n)}/{V(n)})V(n)^{r(n)})$
, so that
$\Delta U(N) = ({\Delta V(N)}/$
${V(N)})V (N)^{r(N)}>0$
.
We can verify that equation (1.15) holds. Indeed,
Additionally, it follows from Remark 1.7 that
$\lim _{N\to \infty }({\Delta U(N)}/{U(N)})=0$
. To check that
$U(N)$
tends to
$\infty $
as
$N\to \infty $
, note that
and so
$\Delta U(N) = V(N)^{r(N)} ({\Delta V(N)}/{V(N)}) = ({\Delta (V(N)^{r(N)})}/{r(N)})\cdot (1+o_{N\to \infty }(1))$
. Similarly,
by our assumptions on r. Let
$T(N) = {V(N)^{r(N)}}/{r(N)}$
, which increases to
$\infty $
. We have shown that
$\lim _{N\to \infty }({\Delta T(N)}/{\Delta U(N)})=1$
and hence
$\lim _{N\to \infty }({T(N)}/ {U(N)}) =1$
. It follows that
$U(N)$
tends to
$\infty $
as
$N\to \infty $
. Since U increases to
$\infty $
and satisfies (1.15), we have by hypothesis that
$\lim _{N\to \infty } {\mathbb {E}}_{n\leqslant N}^Ux_n =L$
. Since
$\lim _{N\to \infty }({T(N)}/{U(N)})=1$
and
$\lim _{n\to \infty }({\Delta T(n)}/{\Delta U(n)})=1$
, we also have that
$\lim _{N\to \infty } {\mathbb {E}}_{n\leqslant N}^{T}x_n =L$
.
We now define a function
$\tilde {U}$
which satisfies the conditions of Lemma 3.2. Let
$\tilde {U}(1) = U(1)$
and
$\tilde {U}(N+1) = \alpha _N \tilde {U}(N)$
, where
and Z is a function which decreases to
$0$
sufficiently slowly. Then
It follows that
We now check that
$\Delta \tilde {U}$
is non-decreasing:
This is non-negative so long as
$\alpha _{N}\geqslant {\Delta \tilde {U}(N)}/{\tilde {U}(N)}+\alpha _{N-1}{\tilde {U}(N-1)}/{\tilde {U}(N)}$
. Using the fact that
${\tilde {U}(N-1)}/{\tilde {U}(N)} = {1}/{\alpha _{N-1}}$
and
${\Delta \tilde {U}(N)}/{\tilde {U}(N)}=1-({1}/{\alpha _{N-1}})$
, this inequality becomes
We can note that
and so it suffices to show that
$({\Delta T(N+1)}/{T(N+1)})(1+Z(N))\geqslant ({\Delta T(N)}/ {T(N)})(1+Z(N-1))$
. However, after rearranging, this is
Since
$\Delta{\log}(V(N))$
is decreasing, this inequality is satisfied so long as Z decreases to
$0$
slowly enough. Thus,
$\Delta \tilde {U}$
is non-decreasing.
Lastly, to show that
$\lim _{N\to \infty } {\mathbb {E}}_{n\leqslant N}^{\tilde {U}}x_n = L$
consider [Reference Kuipers and Niederreiter6, Lemma 7.1], which says that if
$\lim _{N\to \infty } {\mathbb {E}}_{n\leqslant N}^{{T}}x_n {\kern-1pt}={\kern-1pt} L$
,
${\Delta \tilde {U}(N{\kern-1.5pt}+{\kern-1.5pt}1)}/{\Delta \tilde {U}(N)}{\kern-1.5pt}\geqslant{\kern-1.5pt} {\Delta T(N{\kern-1.5pt}+{\kern-1.5pt}1)}/{\Delta T(N)}$
, and there is an
${H{\kern-1.5pt}>{\kern-1.5pt}0}$
such that
${\Delta \tilde {U}(N)}/{\tilde {U}(N)}\leqslant H\cdot ({\Delta T(N)}/{T(N)})$
for all N, then
$\lim _{N\to \infty } {\mathbb {E}}_{n\leqslant N}^{\tilde {U}}x_n = L$
. Using (3.7), we can find an
$H>0$
such that
${\Delta \tilde {U}(N)}/{\tilde {U}(N)}\leqslant H\cdot ({\Delta T(N)}/{T(N)})$
for all N. Also, the inequality
${\Delta \tilde {U}(N+1)}/{\Delta \tilde {U}(N)}\geqslant {\Delta T(N+1)}/{\Delta T(N)}$
holds since
and
So it suffices to show that
$({\alpha _N-1})/({\alpha _{N-1}-1})\geqslant 1+o_{N\to \infty }(1)$
, but we can rearrange (3.8) to see that
$({\alpha _N-1})/({\alpha _{N-1}-1})\geqslant {1}/{\alpha _{N-1}}=1+o_{N\to \infty }(1)$
. This completes the proof.
4
$(4)\implies (5)$
In this section we prove the implication
$(4)\implies (5)$
in Theorem B. As in the previous section, the proof of this implication relies heavily on the Silverman–Toeplitz theorem.
Theorem 4.1. Let Y be a Banach space, let
$(x_n)_{n\in \mathbb {N}}\subseteq Y$
be a bounded sequence, and let
$L\in Y$
. Let
$V:\mathbb {N}\rightarrow (0,\infty )$
be a function which increases to
$\infty $
with
$\Delta{\log}(V(N))$
decreasing to
$0$
such that
$\lim _{N\to \infty }({\Delta{\log}(V(N))}/{\Delta{\log}(N)})=\infty $
. Suppose that
$\lim _{N\to \infty } {\mathbb {E}}_{n\in [N-s(N),N]}x_n = L$
for each non-decreasing function
$s:\mathbb {N}\rightarrow \mathbb {N}$
satisfying
and
$s(N)\leqslant N-1$
for all
$N\in \mathbb {N}$
. Then
$ {\mathbb {E}}_{\mathrm {unif}}^{\log V}x_n = L$
.
Proof. Let
$W(N) = \log (V(N))$
for
$N\in \mathbb {N}$
. Let
$A = A(N)$
and
$B = B(N)$
be arbitrary integer-valued functions with
$W(B)-W(A)\to \infty $
as
$N\to \infty $
. We will show that
$\lim _{N\to \infty } {\mathbb {E}}_{n\in [A,B]}^Wx_n = L$
. Since A and B are arbitrary, this is sufficient to show that
$ {\mathbb {E}}_{\mathrm {unif}}^{\log V}x_n =L$
.
Pick a non-decreasing function
$s\colon \mathbb {N}\rightarrow \mathbb {N}$
satisfying
$\lim _{N\to \infty }s(N)\cdot \Delta W(N) = \infty $
, but which is slow enough such that
$s(N)\leqslant N-1$
for all
$N\in \mathbb {N}$
and
$\lim _{N\to \infty } (({s(B)\cdot \Delta W(B)})/ ({W(B)-W(A)})) = 0$
. Further, we can assume that
$\Delta s(N)\in \{0,1\}$
for all
$N\in \mathbb {N}$
, since the condition
$\lim _{N\to \infty }({\Delta{\log}(V(N))}/{\Delta{\log}(N)})=\lim _{N\to \infty }N\cdot \Delta{\log}(V(N))=\infty $
implies that we can take
$s(N)$
to be asymptotically slower than N. By assumption, we have that
$\lim _{N\to \infty } {\mathbb {E}}_{n\in [N-s(N),N]}x_n=L$
.
We will show that there exists a doubly indexed sequence of non-negative constants
$(c_{N,n})_{N,n\in \mathbb {N}}$
such that
$\lim _{N\to \infty }\sum _{n\in \mathbb {N}}c_{N,n}=1$
,
$\lim _{N\to \infty }c_{N,n_0}=0$
for each fixed
$n_0\in \mathbb {N}$
, and
Once we have shown these properties, we can conclude that
$\lim _{N\to \infty } {\mathbb {E}}_{n\in [A,B]}^Wx_n = L$
by the Silverman–Toeplitz theorem and our assumption that
$\lim _{k\to \infty } {\mathbb {E}}_{n\in [k-s(k),k]}x_n=L$
.
We will define
$(c_{N,n})$
inductively. For
$N\in \mathbb {N}$
, define
$c_{N,n} = 0$
if
$n>B$
or
$n<A$
. Then put
$c_{N,B} = ({s(B)\Delta W(B)}/({W(B)-W(A)}))$
. For
$n\in [A,B]$
, having already defined
$c_{N,n+1},\ldots , c_{N,B}$
, let
With this definition, it is clear that for each
$N\in \mathbb {N}$
we have
Note that
$\lim _{N\to \infty }c_{N,n_0}=0$
for each fixed
$n_0\in \mathbb {N}$
, since
$\lim _{N\to \infty }A = \infty $
and
$c_{N,n_0}=0$
for
$n_0<A$
. Next, we will show that
$c_{N,n}\geqslant 0$
for all n. We proceed with an inductive argument. First note that
$c_{N,B} = ({s(B)\cdot \Delta W(B)})/({W(B)-W(A)})\geqslant 0$
and that
$c_{N,n}=~0$
for
$n>B$
or
$n<A$
. Now suppose that
$A\leqslant k<B$
and
$c_{N,k+1},\ldots , c_{N,B}\geqslant ~0$
. We know that
$\Delta W(k)\geqslant \Delta W(k+1)$
by our assumption that
$\Delta{\log}(V(N))$
is decreasing and so
Let
$z\leqslant B$
be the largest integer such that
$k\geqslant z-s(z)$
, so that
is equal to either
${c_{N,k}}/{s(k)}-{c_{N,z+1}}/{s(z+1)}$
or
${c_{N,k}}/{s(k)}-{c_{N,z+1}}/{s(z+1)}-{c_{N,z+2}}/ {s(z+2)}$
. In either case, we have that
By assumption,
$c_{N,z+1}\geqslant 0$
and so
$c_{N,k}\geqslant 0$
, which completes the induction.
To prove that equation (4.2) holds, observe that
Switching the order of summation, we have
Now we can split the outer sum into two sums in order to apply equation (4.4):
Next, we will show that
$\sum _{n=A-s(A)}^{A-1}x_n\sum _{k= n}^B({c_{N,k}}/{s(k)})\cdot 1_{\{n\geqslant k-s(k)\}}=o_{N\to \infty }(1)$
, which will prove equation (4.2). First, we will use the triangle inequality,
Recall that
$c_{N,k}=0$
for
$k<A$
and so for
$n<A$
we have
Then
By (4.4),
So altogether, we have shown that
Finally, taking
$x_n = 1$
in equation (4.2) shows that
$\sum _{n\in \mathbb {N}}c_{N,n}= 1+o_{N\to \infty }(1)$
, which concludes the proof.
5
$(5)\implies (2)$
In this section we prove the implication
$(5)\implies (2)$
in Theorem B. Our strategy is to emulate the proof of a classical theorem of Schatte.
Theorem 5.1. [Reference Schatte9, Theorem B]
Let
$(x_n)_{n\in \mathbb {N}}$
be a bounded sequence of complex numbers and let
$L\in \mathbb {C}$
. Then
$ {\mathbb {E}}(\infty )(x_n) = L$
if and only if
$ {\mathbb {E}}_{ \mathrm{unif}}^{\log }(x_n) = L$
.
Example 5.2. It is known that
$\lim _{N\to \infty } {\mathbb {E}}_{n\leqslant N}e^{2\pi i \log (n)}=({1}/{N})\sum _{n=1}^Ne^{2\pi i \log (n)}$
does not tend toward
$0$
as
$N\to \infty $
. Indeed,
The sum
$({1}/{N})\sum _{n=1}^Ne^{2\pi i \log (n/N)}$
is a Riemann sum for the integral
$\int _0^1e^{2\pi i \log (x)}dx$
and so
$({1}/{N})\sum _{n=1}^Ne^{2\pi i \log (n/N)} = \int _0^1e^{2\pi i \log (x)}dx + o_{N\to \infty }(1)$
.
Let
$x_n = e^{2\pi i \log (n)}$
and let
$C = \int _0^1e^{2\pi i \log (x)}dx = {1}/({1+2\pi i })$
. Note that
$\|x_n\| = 1$
for all
$n\in \mathbb {N}$
and
$|C|<1$
. Then
In general,
$ {\mathbb {E}}_{n\leqslant N}(k)x_n = C^k\cdot x_N+o_{N\to \infty }(1)$
. From this we can see that as
$k\to \infty $
,
$ {\mathbb {E}}_{n\leqslant N}(k)x_n\to 0$
. So
$ {\mathbb {E}}(\infty )x_n = 0$
. According to Theorem 5.1, this implies that
For our purposes, we need to begin with the following theorem, which is a straightforward extension of [Reference Schatte8, Satz 1].
Theorem 5.3. Let Y be a Banach space, let
$(x_n)_{n\in \mathbb {N}}\subseteq Y$
be a bounded sequence. Let
$V:\mathbb {N}\rightarrow (0,\infty )$
be a function which increases to
$\infty $
and satisfies
$\lim _{N\to \infty }({\Delta V(N)}/ {V(N)}) =0$
. Then for each
$k\in \mathbb {N}$
,
Proof. We proceed by induction on k. There is nothing to prove when
$k=0$
, so suppose that equation (5.1) holds and we will show that
Then
We will show that
from which equation (5.2) follows. Recall from Remark 1.7 that
${\Delta V(N)}/{V(N)} = \Delta{\log}(V(N))\cdot (1+o_{N\to \infty }(1))$
. Then
Let
$\ell _n = \log ({V(N)}/{V(n)})$
and
$G_n = \sum _{i=m}^n\Delta{\log}(V(i))$
, so that
$\Delta G_n = \Delta {\log} (V(n)) = -\Delta \ell _n$
. Then using summation by parts, we have
The first two terms are equal to
$0$
since
$\ell _N = 0$
and
$G_{m-1}=0$
. Next, noting that
$\lim _{n\to \infty }({\ell _{n}}/{\ell _{n+1}}) = 1$
, we have
Additionally, we can write
So,
Comparing with (5.3), we have
so
as desired.
Lemma 5.4. Let Y be a Banach space and let
$(x_n)_{n\in \mathbb {N}}\subseteq Y$
be a bounded sequence. Let
$V:\mathbb {N}\rightarrow (0,\infty )$
be a function which increases to
$\infty $
and satisfies
$\lim _{N\to \infty }({\Delta V(N)}/ {V(N)}) =0$
. Suppose that
Then
$ {\mathbb {E}}(\infty )^Vx_n = 0$
.
Proof. Pick
$k\in \mathbb {N}$
arbitrarily large and define
$F(x) = ({x}/{k!})\log ({1}/{x})^k$
, with
$F(0)=0$
. Note that F is continuous and F is increasing on the interval
$(0,e^{-k})$
and decreasing on the interval
$(e^{-k},1)$
. From Theorem 5.3, we have that
Summation by parts gives us that
Pick
$C>0$
such that
$\limsup _{N\to \infty }|\!\sum _{n=1}^N({\Delta V(n)}/{V(n)})x_n|\leqslant C$
. Then
The result follows from the fact that
$\lim _{k\to \infty }F(e^{-k})=0$
.
Theorem 5.5. Let Y be a Banach space and let
$(x_n)_{n\in \mathbb {N}}\subseteq Y$
be a bounded sequence. Let
$V:\mathbb {N}\rightarrow (0,\infty )$
be a function which increases to
$\infty $
and satisfies
$\lim _{N\to \infty }({\Delta V(N)}/{V(N)})=0$
. Suppose that
$ {\mathbb {E}}_{\mathrm {unif}}^{\log V}(x_n) = L$
. Then
$ {\mathbb {E}}(\infty )^Vx_n = L$
.
Proof. Without loss of generality, assume
$L=0$
. Pick any
$\varepsilon>0$
. We will find bounded sequences
$(y_n)_{n\in \mathbb {N}}, (z_n)_{n\in \mathbb {N}}$
with
$x_n=y_n+z_n$
for each
$n\in \mathbb {N}$
, such that
$\sup _{N\in \mathbb {N}}|y_N|<\varepsilon $
and
$\limsup _{N\to \infty }|\!\sum _{n=1}^N({\Delta V(n)}/{V(n)})z_n|<\infty $
. From Lemma 5.4 it will follow that
which will conclude the proof.
Recall from Remark 1.7 that
${\Delta V(n)}/{V(n)} = \Delta{\log}(V(n))\cdot (1+o_{n\to \infty }(1))$
. We know that
tends to
$0$
whenever
$\log (V(B))-\log (V(A))$
tends to
$\infty $
, and so it follows that we also have that
tends to L whenever
$\log (V(B))-\log (V(A))$
tends to
$\infty $
.
We can find a
$K_0>0$
such that for any
$M,N$
with
$W(N)-W(M)>K_0$
,
Define a sequence
$(N_i)$
as follows. Let
$N_1=1$
and, having picked
$N_i$
for
$i\in \mathbb {N}$
, pick the smallest
$N_{i+1}>N_i$
such that
$\log (V(N_{i+1}-1))-\log (V(N_i))>K_0$
. For each
$i\in \mathbb {N}$
, define
so that the sequence
$(y_k)_{k\in \mathbb {N}}$
is constant on each interval
$[N_i,N_{i+1}-1]$
and
$|y_n|<\varepsilon $
for all
$n\in \mathbb {N}$
. Put
$z_n = x_n-y_n$
for all
$n\in \mathbb {N}$
, and note that
$(z_n)$
is bounded since
$(x_n)_{n\in \mathbb {N}}$
and
$(y_n)_{n\in \mathbb {N}}$
are bounded. All that is left is to show that
$\limsup _{N\to \infty }|\!\sum _{n=1}^N({\Delta V(n)}/{V(n)})z_n|<\infty $
.
Pick
$N\in \mathbb {N}$
and pick i such that
$N_{i-1}<N\leqslant N_{i}$
. Then
We have that
$z_k = x_k-y_k$
, and
$y_k$
is constant for
$k\in [N_{m-1},N_m-1]$
, so
Then
We have that
$\sum _{k=N+1}^{N_i}({\Delta V(k)}/{V(k)})\leqslant \sum _{k=N_{i-1}}^{N_i}({\Delta V(k)}/{V(k)})$
which is close to
$\log (V(N_{i}))-\log (V(N_{i-1}+1))$
. By definition of
$(N_i)_{i\in \mathbb {N}}$
, we have that
$\log (V(N_{i}))-\log (V(N_{i-1}+1))\leqslant K_0+1$
for all large enough i. So we have
$\limsup _{N\to \infty } |\!\sum _{n=1}^N({\Delta V(n)}/{V(n)})z_n|<\infty $
as desired.
Acknowledgements
The creation of this paper would not have been possible without the generous advice and direction from the author’s advisor, Vitaly Bergelson. We would also like to thank Nikos Frantzikinakis for helpful conversation, Saúl Rodríguez Martín for the inspiration of Example 5.2 and for finding mistakes in an earlier version of this paper, and an anonymous referee for helpful comments.
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