# Counterexamples to generalizations of the Erdős $B+B+t$ problem

Ethan Ackelsberg  Institute of Mathematics, École Polytechnique Fédérale de
Lausanne, Lausanne, Switzerland<EMAIL_ADDRESS>
###### Abstract.

Following their resolution of the Erdős $B+B+t$ problem in [10], Kra Moreira,
Richter, and Robertson posed a number of questions and conjectures related to
infinite configurations in positive density subsets of the integers and other
amenable groups in [11]. We give a negative answer to several of the questions
and conjectures in [11] by producing families of counterexamples based on a
construction of Ernst Straus.

Included among our counterexamples, we exhibit, for any $\varepsilon>0$, a set
$A\subseteq\mathbb{N}$ with multiplicative upper Banach density at least
$1-\varepsilon$ such that $A$ does not contain any dilated product set
$\\{b_{1}b_{2}t:b_{1},b_{2}\in B,b_{1}\neq b_{2}\\}$ for an infinite set
$B\subseteq\mathbb{N}$ and $t\in\mathbb{Q}_{>0}$. We also prove the existence
of a set $A\subseteq\mathbb{N}$ with additive upper Banach density at least
$1-\varepsilon$ such that $A$ does not contain any polynomial configuration
$\\{b_{1}^{2}+b_{2}+t:b_{1},b_{2}\in B,b_{1}<b_{2}\\}$ for an infinite set
$B\subseteq\mathbb{N}$ and $t\in\mathbb{Z}$. Counterexamples to some closely
related problems are also discussed.

###### 1991 Mathematics Subject Classification:

Primary: 05D10; Secondary: 11B13, 11B30

##  1\. Introduction

In [10], Kra, Moreira, Richter, and Robertson provided a positive resolution
to the Erdős $B+B+t$ problem (asked in [5, 6, 7]) by showing that any positive
density subset of the integers contains a shift of an infinite sumset, i.e. a
configuration of the form $B\oplus B+t=\\{b_{1}+b_{2}+t:b_{1},b_{2}\in
B,b_{1}\neq b_{2}\\}$ for some infinite set $B\subseteq\mathbb{N}$ and some
$t\in\mathbb{Z}$ (and, moreover, $B$ may be taken as a subset of $A$). One is
then left with the natural question of which other infinite combinatorial
configurations can be found in sets of positive density of the integers or of
other amenable groups. The survey article [11] provides an extensive overview
of questions along these lines, with some partial results in both the positive
and negative direction. The goal of the present paper is to provide a negative
answer to several of the questions and conjectures posed in [11].

###  1.1. Sumsets in abelian groups

First, we address the generalization of the Erdős $B+B+t$ problem to abelian
groups. Let $\Gamma$ be a countable discrete abelian group. A _Følner
sequence_ in $\Gamma$ is a sequence of finite subsets
$(\Phi_{N})_{N\in\mathbb{N}}$ of $\Gamma$ such that for any $x\in\Gamma$,

$\lim_{N\to\infty}\frac{\left|(\Phi_{N}+x)\triangle\Phi_{N}\right|}{|\Phi_{N}|}=0.$

The _upper and lower density of a subset $A\subseteq\Gamma$ along a Følner
sequence $\Phi=(\Phi_{N})_{N\in\mathbb{N}}$_ are given by

$\overline{d}_{\Phi}(A)=\limsup_{N\to\infty}\frac{\left|A\cap\Phi_{N}\right|}{|\Phi_{N}|}\qquad\text{and}\qquad\underline{d}_{\Phi}(A)=\liminf_{N\to\infty}\frac{\left|A\cap\Phi_{N}\right|}{|\Phi_{N}|}$

The _upper and lower Banach density_ of $A\subseteq\Gamma$ are defined by
$d^{*}(A)=\sup_{\Phi}\overline{d}_{\Phi}(A)$ and
$d_{*}(A)=\inf_{\Phi}\underline{d}_{\Phi}(A)$, where the supremum and infimum
are taken over all Følner sequences $\Phi$ in $\Gamma$. In [4], Charamaras and
Mountakis prove a generalization of the main result of [10] for abelian groups
whose “even” elements form a finite index subgroup:

######  Theorem 1.1 ([4, Corollary 1.12]111The main result of [4] applies to a
class of amenable groups under a technical assumption on the set of squares.
The result stated in Theorem 1.1 is the special case of their result in the
setting of abelian groups.).

Let $\Gamma$ be a countable discrete abelian group such that
$2\Gamma=\\{2x:x\in\Gamma\\}$ is a finite index subgroup of $\Gamma$. If
$A\subseteq\Gamma$ is a set of positive upper Banach density, then there
exists an infinite set $B\subseteq\Gamma$ and a shift $t\in\Gamma$ such that

$B\oplus B+t=\\{b_{1}+b_{2}+t:b_{1},b_{2}\in B,b_{1}\neq b_{2}\\}\subseteq A.$

We prove that the Erdős $B+B+t$ problem for an abelian group $\Gamma$ has a
negative answer whenever $2\Gamma$ is of infinite index, disproving [11,
Conjecture 5.14]222Terence Tao has also observed that the condition
$[\Gamma:2\Gamma]<\infty$ in Theorem 1.1 is necessary (see [13])..

######  Theorem 1.2.

Suppose $\Gamma$ is a countable discrete abelian group and $2\Gamma$ is a
subgroup of infinite index. Then for any $\varepsilon>0$, there exists
$A\subseteq\Gamma$ such that $d^{*}(A)>1-\varepsilon$ and if
$B\subseteq\Gamma$ and $t\in\Gamma$ with $B\oplus B+t\subseteq A$, then $B$ is
finite.

In fact, we can prove a similar result for higher-order sumsets as well. For
$k\in\mathbb{N}$ and $B\subseteq\Gamma$, let

$B^{\oplus k}=\left\\{\sum_{x\in F}x:F\subseteq B,|F|=k\right\\}.$

######  Theorem 1.3.

Let $k\in\mathbb{N}$. Suppose $\Gamma$ is a countable discrete abelian group
and $k\Gamma$ is a subgroup of infinite index. Then for any $\varepsilon>0$,
there exists $A\subseteq\Gamma$ such that $d^{*}(A)>1-\varepsilon$ and if
$B\subseteq\Gamma$ and $t\in\Gamma$ with $B^{\oplus k}+t\subseteq A$, then $B$
is finite.

###  1.2. Product sets in $(\mathbb{N},\times)$

The semigroup $(\mathbb{N},\times)$ of positive integers under multiplication
has the property that the subsemigroup of $k$th powers,
$\\{n^{k}:n\in\mathbb{N}\\}$, has zero (multiplicative) upper Banach density
for every $k\geq 2$. With a small amount of additional work, we can use
Theorem 1.3 to construct a counterexample in $(\mathbb{N},\times)$ for all
$k\geq 2$ simultaneously:

######  Theorem 1.4.

Let $\varepsilon>0$. There exists $A\subseteq\mathbb{N}$ with
$d_{\times}^{*}(A)>1-\varepsilon$ such that if $B\subseteq\mathbb{N}$,
$t\in\mathbb{Q}_{>0}$, and $k\geq 2$ with

$tB^{\odot k}=\left\\{t\prod_{i=1}^{k}b_{i}:b_{1},\dots,b_{k}\in
B,b_{1}<\dots<b_{k}\right\\}\subseteq A,$

then $B$ is finite.

The case $k=2$ in Theorem 1.4 gives a disproof of [11, Conjecture 5.1].

###  1.3. Polynomial configurations

Our next counterexample deals with polynomial patterns in the integers.
Seeking a sumset analogue of the Furstenberg–Sárközy theorem [9, 12], Kra,
Moreira, Richter, and Robertson asked whether every set of positive density in
the integers contains a configuration of the form

$\\{b_{1}^{2}+b_{2}:b_{1},b_{2}\in B,b_{1}<b_{2}\\}+t$

for some infinite set $B\subseteq\mathbb{N}$ and some $t\in\mathbb{Z}$ (see
[11, Question 3.13]). We show that the answer is negative and remains negative
for any polynomial of degree at least two in place of the squares:

######  Theorem 1.5.

Let $P(x)\in\mathbb{Q}[x]$ be an integer-valued polynomial333By an _integer-
valued polynomial_ , we mean a polynomial taking integer values on the
integers. That is, $P(\mathbb{Z})\subseteq\mathbb{Z}$. with $\deg{P}\geq 2$.
Then for any $\varepsilon>0$, there exists a set $A\subseteq\mathbb{N}$ with
$d^{*}(A)>1-\varepsilon$ such that if $B\subseteq\mathbb{N}$,
$t\in\mathbb{Z}$, and

$P(B)\footnotesize~{}\rotatebox[origin={c}]{90.0}{$\triangleplus$}~{}B+t=\left\\{P(b_{1})+b_{2}+t:b_{1},b_{2}\in
B,b_{1}<b_{2}\right\\}\subseteq A,$

then $B$ is finite.

######  Remark 1.6.

Let us comment briefly on the choice of ordering for the configuration
appearing in Theorem 1.5. One could also consider the related configuration
$P(B)\footnotesize~{}\rotatebox[origin={c}]{270.0}{$\triangleplus$}~{}B=\\{P(b_{1})+b_{2}:b_{1},b_{2}\in
B,b_{1}>b_{2}\\}$. In [11, Example 3.14], a counterexample of an even stronger
form is given for this configuration. Namely, there exists a set
$A\subseteq\mathbb{N}$ with $d(A)=1$ such that $A$ does not contain any set of
the form
$B^{2}\footnotesize~{}\rotatebox[origin={c}]{270.0}{$\triangleplus$}~{}C=\\{b^{2}+c:b\in
B,c\in C,b>c\\}$ with $B,C\subseteq\mathbb{N}$ infinite, and the only property
of the squares used for the proof is that the set of squares
$\\{n^{2}:n\in\mathbb{N}\\}$ is of zero density, so the argument applies
equally well to any polynomial of degree at least 2.

The configuration
$P(B)\footnotesize~{}\rotatebox[origin={c}]{90.0}{$\triangleplus$}~{}B$ in
Theorem 1.5 is better behaved in several respects. First, if
$A\subseteq\mathbb{N}$ and $d^{*}(A)=1$, then $A$ contains
$P(B)\footnotesize~{}\rotatebox[origin={c}]{90.0}{$\triangleplus$}~{}B$ for
some infinite set $B\subseteq\mathbb{N}$. Second, if $A\subseteq\mathbb{N}$
and $d^{*}(A)>0$, then there are infinite sets $B,C\subseteq\mathbb{N}$ such
that
$P(B)\footnotesize~{}\rotatebox[origin={c}]{90.0}{$\triangleplus$}~{}C\subseteq
A$, and, moreover, one may choose $C\subseteq A-P(0)$; see [11, Corollary
3.33].

Theorem 1.5 shows that, despite the improved behavior of ordered sumsets with
the polynomial input being of smaller size, once may still avoid infinite
configurations of the form
$P(B)\footnotesize~{}\rotatebox[origin={c}]{90.0}{$\triangleplus$}~{}B$ in all
shifts of a set of positive density. We should note that our proof makes use
of the algebraic structure of polynomials in addition to the fact that
$P(\mathbb{N})$ has zero density. A natural lingering question is whether one
can construct counterexamples without this algebraic structure; see Question
5.2 and the accompanying discussion at the end of the paper.

###  1.4. Product-sum sets

Finally, we show that [11, Question 3.18], which is a variant of [11, Question
3.13], also has a negative answer:

######  Theorem 1.7.

For any $\varepsilon>0$, there exists a set $A\subseteq\mathbb{N}$ with
$d^{*}(A)>1-\varepsilon$ such that if $B\subseteq\mathbb{N}$,
$t\in\mathbb{Z}$, and

$B\footnotesize~{}\rotatebox[origin={c}]{90.0}{$\trianglecdot$}~{}B\footnotesize~{}\rotatebox[origin={c}]{90.0}{$\triangleplus$}~{}B+t=\left\\{b_{1}\cdot
b_{2}+b_{3}+t:b_{1},b_{2},b_{3}\in B,b_{1}<b_{2}<b_{3}\right\\}\subseteq A,$

then $B$ is finite.

###  1.5. Remarks about partition regularity versus density regularity

Consider the following families of sets discussed in the above results:

  * •

For an abelian group $\Gamma$:
$\mathscr{C}_{k\textup{-sumset}}=\\{A\subseteq\Gamma:\exists
B\subseteq\Gamma~{}\text{infinite},B^{\oplus k}\subseteq A\\}$.

  * •

$\mathscr{C}_{k\textup{-product set}}=\left\\{A\subseteq\mathbb{N}:\exists
B\subseteq\mathbb{N}~{}\text{infinite},B^{\odot k}\subseteq A\right\\}$ and
$\mathscr{C}_{\textup{product}}=\bigcup_{k\geq
2}\mathscr{C}_{k\textup{-product set}}$.

  * •

For an integer-valued polynomial $P(x)\in\mathbb{Q}[x]$:
$\mathscr{C}_{P}=\left\\{A\subseteq\mathbb{N}:\exists
B\subseteq\mathbb{N}~{}\text{infinite},P(B)\footnotesize~{}\rotatebox[origin={c}]{90.0}{$\triangleplus$}~{}B\subseteq
A\right\\}$.

  * •

$\mathscr{C}_{\times,+}=\\{A\subseteq\mathbb{N}:\exists
B\subseteq\mathbb{N}~{}\text{infinite},B\footnotesize~{}\rotatebox[origin={c}]{90.0}{$\trianglecdot$}~{}B\footnotesize~{}\rotatebox[origin={c}]{90.0}{$\triangleplus$}~{}B\subseteq
A\\}$.

Theorems 1.3, 1.4, 1.5, and 1.7 provide examples of sets with density
arbitrarily close to 1 not containing any shift of an element of
$\mathscr{C}_{k\textup{-sumset}}$, $\mathscr{C}_{\textup{product}}$,
$\mathscr{C}_{P}$, or $\mathscr{C}_{\times,+}$, respectively. We can therefore
rephrase the main theorems as showing that these families of configuration
together with their shifts are not _density regular_. This is made more
meaningful by the following observation, which is a simple consequence of
Ramsey’s theorem.

######  Proposition 1.8.

The families $\mathscr{C}_{k\textup{-sumset}}$,
$\mathscr{C}_{\textup{product}}$, $\mathscr{C}_{P}$, and
$\mathscr{C}_{\times,+}$ are partition regular. That is, for each
$\mathscr{C}\in\left\\{\mathscr{C}_{k\textup{-sumset}},\mathscr{C}_{\textup{product}},\mathscr{C}_{P},\mathscr{C}_{\times,+}\right\\}$,
if $C\in\mathscr{C}$ and $C=\bigcup_{i=1}^{r}C_{i}$, then
$C_{i}\in\mathscr{C}$ for some $i\in\\{1,\dots,r\\}$.

######  Remark 1.9.

One can check that each of the families $\mathscr{C}_{k\textup{-sumset}}$,
$\mathscr{C}_{\textup{product}}$, $\mathscr{C}_{P}$, and
$\mathscr{C}_{\times,+}$ contains all thick sets444A set is _thick_ if it
contains a shift of every finite set, or, equivalently, if it has upper Banach
density equal to 1. and deduce that every piecewise syndetic set555A set is
_syndetic_ if it has non-empty intersection with every thick set, or,
equivalently, if it has positive lower Banach density. A _piecewise syndetic
set_ is an intersection of a syndetic set with a thick set. The family of
piecewise syndetic sets is partition regular. contains a shift of each of the
configurations we have considered. The counterexamples we produce below will
therefore be examples of non-piecewise syndetic sets with positive density.
Such sets have been explored previously in [3, 2].

##  2\. Revisiting the Straus example

All of the constructions in this paper are based on Ernst Straus’s
counterexample to a density version of Hindman’s theorem:

######  Theorem 2.1 (Ernst Straus, cf. [1, Theorem 2.20]).

Let $\varepsilon>0$. There is a set $A\subseteq\mathbb{N}$ with
$d^{*}(A)>1-\varepsilon$ such that $A$ does not contain any configuration of
the form

$FS\left((x_{n})_{n\in\mathbb{N}}\right)+t=\left\\{x_{i_{1}}+\dots+x_{i_{k}}+t:k\in\mathbb{N},i_{1}<\dots<i_{k}\right\\}.$

for an infinite sequence $x_{1}<x_{2}<\dots\in\mathbb{N}$ and
$t\in\mathbb{Z}$.

Our goal is prove a similar result for the combinatorial configurations
mentioned in the introduction, and the basic approach will be very similar. In
order to motivate the new constructions in this paper, let us briefly review
the main ideas in the proof of Theorem 2.1.

The first observation to make is that IP-sets (sets of the form
$FS((x_{n})_{n\in\mathbb{N}})$ for an infinite sequence
$x_{1}<x_{2}<\dots\in\mathbb{N}$) are “divisible” in the following sense: if
$A$ contains an IP-set, then $A\cap k\mathbb{N}$ contains an IP-set for every
$k\in\mathbb{N}$. The idea to prove Theorem 2.1 is then to remove from
$\mathbb{N}$ an infinite arithmetic progression $k_{t}\mathbb{N}+t$ for each
$t\in\mathbb{Z}$ with $k_{t}$ growing sufficiently quickly so that
$\bigcup_{t\in\mathbb{Z}}(k_{t}\mathbb{N}+t)$ has small density. Isolating the
essential properties utilized by Straus, we can prove the following general
result:

######  Theorem 2.2.

Let $\Gamma$ be an abelian group. Let $\mathscr{C}$ be a family of infinite
subsets of $\Gamma$, and suppose there is a family $\mathscr{D}$ of subsets of
$\Gamma$ and a Følner sequence $\Phi$ with the following properties:

  1. (1)

For any $C\in\mathscr{C}$ and $D\in\mathscr{D}$, one has $C\cap
D\in\mathscr{C}$, and

  2. (2)

$\inf_{D\in\mathscr{D}}\overline{d}_{\Phi}(D)=0$.

Then for any $\varepsilon>0$, there exists $A\subseteq\Gamma$ with
$\underline{d}_{\Phi}(A)>1-\varepsilon$ such that for any $t\in\Gamma$,
$A-t\notin\mathscr{C}$.

Taking $\mathscr{C}=\\{A\subseteq\mathbb{N}:A~{}\text{contains an IP-set}\\}$
and $\mathscr{D}=\\{k\mathbb{N}:k\in\mathbb{N}\\}$ reproduces Theorem 2.1.

##  3\. Key lemma and proof of Theorem 2.2

The key lemma for proving Theorems 2.2 can be interpreted as an approximate
version of countable subadditivity for the upper density along a Følner
sequence:

######  Lemma 3.1.

Let $\Gamma$ be a countable discrete abelian group with a Følner sequence
$\Phi$. Let $(A_{n})_{n\in\mathbb{N}}$ be a countable family of subsets of
$\Gamma$. There exists $A\subseteq\Gamma$ such that

  1. (1)

for any $n\in\mathbb{N}$, $A_{n}\setminus A$ is finite, and

  2. (2)

$\overline{d}_{\Phi}(A)\leq\sum_{n\in\mathbb{N}}\overline{d}_{\Phi}(A_{n})$.

###### Proof.

Let $\delta_{n}=\overline{d}_{\Phi}(A_{n})$, and let
$\delta=\sum_{n\in\mathbb{N}}\delta_{n}$. If $\delta\geq 1$, there is nothing
to prove (one may take $A=\Gamma$), so assume $\delta<1$. By the definition of
upper density along $\Phi$, we have

$\frac{|A_{n}\cap\Phi_{k}|}{|\Phi_{k}|}\leq\delta_{n}+\varepsilon(n,k)$

for some $\varepsilon(n,k)\in[0,1]$ with $\varepsilon(n,k)\to 0$ as
$k\to\infty$.

We will construct a sequence $(k_{n})_{n\in\mathbb{N}}$ be induction, put
$A^{\prime}_{n}=A_{n}\setminus\bigcup_{k<k_{n}}\Phi_{k}$, and then let
$A=\bigcup_{n\in\mathbb{N}}A^{\prime}_{n}$. Note that property (1) is
automatically satisfied (regardless of the choice of
$(k_{n})_{n\in\mathbb{N}}$) since $A_{n}\setminus
A\subseteq\bigcup_{k<k_{n}}\Phi_{k}$.

Let $k_{1}=1$. For $n\in\mathbb{N}$, given $k_{1},\dots,k_{n-1}$, choose
$k_{n}>k_{n-1}$ such that

$\sum_{j=1}^{n}\varepsilon(j,k)\leq\frac{1}{n}$

for all $k\geq k_{n}$. We check that property (2) is satisfied for this choice
of $(k_{n})_{n\in\mathbb{N}}$. For $k\in\mathbb{N}$, let
$n_{k}=\max\\{n\in\mathbb{N}:k_{n}\leq k\\}$. Note that
$A\cap\Phi_{k}=\bigcup_{j=1}^{n_{k}}A_{j}^{\prime}\cap\Phi_{k}\subseteq\bigcup_{j=1}^{n_{k}}A_{j}\cap\Phi_{k}$.
Therefore

$\frac{|A\cap\Phi_{k}|}{|\Phi_{k}|}\leq\sum_{j=1}^{n_{k}}\frac{|A_{j}\cap\Phi_{k}|}{|\Phi_{k}|}\leq\sum_{j=1}^{n_{k}}\delta_{j}+\sum_{j=1}^{n_{k}}\varepsilon(j,k)\leq\delta+\frac{1}{n_{k}},$

so

$\overline{d}_{\Phi}(A)\leq\delta+\limsup_{k\to\infty}\frac{1}{n_{k}}=\delta.$

∎

######  Remark 3.2.

The proof of Lemma 3.1 does not use that $\Phi$ is a Følner sequence nor does
it utilize any algebraic structure of $\Gamma$. The same result holds for any
notion of upper density on a set obtained by taking a limit of finitely-
supported probability measures.

We can now give a very short proof of Theorem 2.2.

###### Proof of Theorem 2.2.

Let $\varepsilon>0$. By property (2), we may choose a family of sets
$(D_{t})_{t\in\Gamma}$ in $\mathscr{D}$ such that
$\sum_{t\in\Gamma}\overline{d}_{\Phi}(D_{t})<\varepsilon$. Then by Lemma 3.1,
let $S\subseteq\Gamma$ such that $\overline{d}_{\Phi}(S)<\varepsilon$ and
$(D_{t}+t)\setminus S$ is finite for each $t\in\Gamma$. Let $A=\Gamma\setminus
S$. Then $\underline{d}_{\Phi}(A)=1-\overline{d}_{\Phi}(S)>1-\varepsilon$.
Moreover, for each $t\in\Gamma$, the intersection $(A-t)\cap
D_{t}=D_{t}\setminus(S_{t}-t)$ is finite. In particular, $(A-t)\cap
D_{t}\notin\mathscr{C}$, so by property (1), $A-t\notin\mathscr{C}$. ∎

##  4\. Constructing the counterexamples

Using Theorem 2.2, all that remains in order to prove Theorems 1.3, 1.4, 1.5,
and 1.7 is to find the appropriate family of sets $\mathscr{D}$ for each
corresponding collection of combinatorial configurations $\mathscr{C}$.

###  4.1. Sumsets in abelian groups

Let $\Gamma$ be a countable discrete abelian group. Recall
$\mathscr{C}_{k\textup{-sumset}}=\\{A\subseteq\Gamma:\exists
B\subseteq\Gamma~{}\text{infinite},B^{\oplus k}\subseteq A\\}$.

######  Lemma 4.1.

Let $\Gamma$ be a countable discrete abelian group and $k\in\mathbb{N}$. If
$C\in\mathscr{C}_{k\textup{-sumset}}$ and $\Lambda\leq\Gamma$ is a finite
index subgroup with $k\Gamma\subseteq\Lambda$, then
$C\cap\Lambda\in\mathscr{C}_{k\textup{-sumset}}$.

###### Proof.

Let $B\subseteq\Gamma$ be an infinite set such that $B^{\oplus k}\subseteq C$.
By the pigeonhole principle, there exists $x\in\Gamma$ such that
$B^{\prime}=B\cap(\Lambda+x)$ is infinite. Since $kx\in\Lambda$, we conclude
$B^{\prime\oplus k}\subseteq C\cap\Lambda$. ∎

###### Proof of Theorem 1.3.

Let
$\mathscr{D}=\left\\{\Lambda\leq\Gamma:k\Gamma\subseteq\Lambda~{}\text{and}~{}[\Gamma:\Lambda]<\infty\right\\}$.
A finite index subgroup $\Lambda\leq\Gamma$ has uniform density
$\frac{1}{[\Gamma:\Lambda]}$, so by Theorem 2.2 and Lemma 4.1, it suffices to
show $\sup_{\Lambda\in\mathscr{D}}[\Gamma:\Lambda]=\infty$. By assumption,
$[\Gamma:k\Gamma]=\infty$, so $\Gamma/k\Gamma$ is an infinite group. Since
each element of $\Gamma/k\Gamma$ has order dividing $k$, we may write
$\Gamma/k\Gamma$ as an infinite direct sum of cyclic groups
$\Gamma/k\Gamma=\bigoplus_{i\in\mathbb{N}}\mathbb{Z}/n_{i}\mathbb{Z}$ with
$n_{i}\mid k$ for each $i\in\mathbb{N}$ (see, e.g., [8, Theorem 17.2]). The
subgroups
$\widetilde{\Lambda}_{j}=\left\\{(x_{i})_{i\in\mathbb{N}}:x_{i}=0~{}\text{for}~{}i<j\right\\}\leq\Gamma/k\Gamma$
satisfy $[\Gamma/k\Gamma:\widetilde{\Lambda}_{j}]=\prod_{i<j}n_{i}$, so
$\sup_{j\in\mathbb{N}}[\Gamma/k\Gamma:\widetilde{\Lambda}_{j}]=\infty$.
Lifting $\widetilde{\Lambda}_{j}$ to a subgroup $\Lambda_{j}\leq\Gamma$, we
have $\Lambda_{j}\in\mathscr{D}$ and
$\sup_{j\in\mathbb{N}}[\Gamma:\Lambda_{j}]=\infty$. ∎

###  4.2. Product sets in $(\mathbb{N},\times)$

###### Proof of Theorem 1.4.

Consider the group isomorphism
$\varphi:(\mathbb{Q}_{>0},\times)\to(\bigoplus_{n\in\mathbb{N}}\mathbb{Z},+)$
that maps an element to its prime factorization. That is, if
$q=\prod_{i=1}^{s}p_{i}^{e_{i}}$, where $p_{1}<p_{2}<\dots$ is the increasing
enumeration of the primes, then
$\varphi(q)=(e_{1},e_{2},\dots,e_{s},0,\dots)$.

Let $\Phi$ be a Følner sequence in $(\mathbb{N},\times)$, and let
$\widetilde{\Phi}$ be the Følner sequence in
$(\bigoplus_{n\in\mathbb{N}}\mathbb{Z},+)$ given by
$\widetilde{\Phi}_{N}=\varphi(\Phi_{N})$. By Theorem 1.3, we may find, for
each $k\geq 2$, a set $A_{k}\subseteq\bigoplus_{n\in\mathbb{N}}\mathbb{Z}$
such that $\underline{d}_{\widetilde{\Phi}}(A_{k})>1-2^{-k}\varepsilon$ and no
shift of $A_{k}$ contains an infinite $k$-fold sumset $B^{\oplus k}$. (As
stated in the introduction, Theorem 1.3 only guarantees
$d^{*}(A_{k})>1-2^{-k}\varepsilon$. However, from the conclusion of Theorem
2.2, we see that one may replace the upper Banach density with the lower
density along the Følner sequence $\widetilde{\Phi}$.)

Put $N_{k}=(\bigoplus_{n\in\mathbb{N}}\mathbb{Z})\setminus A_{k}$. Then
$\overline{d}_{\widetilde{\Phi}}(N_{k})<2^{-k}\varepsilon$, so by Lemma 3.1,
there exists a set $N\subseteq\bigoplus_{n\in\mathbb{N}}\mathbb{Z}$ such that
$N_{k}\setminus N$ is finite for each $k\geq 2$ and
$\overline{d}_{\widetilde{\Phi}}(N)<\varepsilon$.

Let $A=\varphi^{-1}\left((\bigoplus_{n\in\mathbb{N}}\mathbb{Z})\setminus
N\right)\cap\mathbb{N}$. Then
$d_{\times}^{*}(A)\geq\underline{d}_{\Phi}(A)=1-\overline{d}_{\widetilde{\Phi}}(N)>1-\varepsilon$.
Suppose $B\subseteq\mathbb{N}$, $t\in\mathbb{Q}_{>0}$, $k\geq 2$, and
$tB^{\odot k}\subseteq A$. Then $\varphi\left(tB^{\odot k}\right)\cap
N=\emptyset$. But $\varphi$ is a group isomorphism, so we have
$\varphi\left(tB^{\odot k}\right)=\varphi(t)+\varphi(B)^{\oplus k}$.
Therefore, from the construction of $N$, the set
$\left(\varphi(t)+\varphi(B)^{\oplus k}\right)\cap N_{k}$ is finite. Hence,
removing finitely elements from $B$, we obtain a cofinite subset
$B^{\prime}\subseteq B$ such that
$\left(\varphi(t)+\varphi(B^{\prime})^{\oplus k}\right)\subseteq A_{k}$. By
construction, the set $A_{k}-\varphi(t)$ does not contain any infinite
$k$-fold subset, so we conclude that $B^{\prime}$ (and hence $B$) is a finite
set. ∎

###  4.3. Polynomial configurations

######  Lemma 4.2.

Let $P(x)\in\mathbb{Q}[x]$ be an integer-valued polynomial. Let
$k\in\mathbb{N}$. If $C\in\mathscr{C}_{P}$, then there exists $n\in\mathbb{N}$
such that $C\cap(k\mathbb{N}+P(n)+n)\in\mathscr{C}_{P}$.

###### Proof.

Let $B\subseteq\mathbb{N}$ be an infinite set such that
$P(B)\footnotesize~{}\rotatebox[origin={c}]{90.0}{$\triangleplus$}~{}B\subseteq
A$. Let $D\in\mathbb{N}$ such that $D\cdot P(x)\in\mathbb{Z}[x]$ has integer
coefficients. Then for any $k\in\mathbb{N}$, if $n\equiv m\pmod{Dk}$, then
$P(n)\equiv P(m)\pmod{k}$. By the pigeonhole principle, there exists
$n\in\\{0,1,\dots,Dk-1\\}$ such that $B^{\prime}=B\cap(Dk\mathbb{N}+n)$ is
infinite. Then
$P(B^{\prime})\footnotesize~{}\rotatebox[origin={c}]{90.0}{$\triangleplus$}~{}B^{\prime}\subseteq
A\cap(k\mathbb{N}+P(n)+n)$. ∎

As suggested by Lemma 4.2, the relevant family $\mathscr{D}$ for applying
Theorem 2.2 will be the family of sets of the form
$D_{k}=\bigcup_{n\in\mathbb{N}}\left(k\mathbb{N}+P(n)+n\right)$ for
$k\in\mathbb{N}$. In order to show that such sets have arbitrarliy small
density (condition (2) in Theorem 2.2), we use the following property of
polynomials:

######  Lemma 4.3.

Let $P(x)\in\mathbb{Q}[x]$ be an integer-valued polynomial with $\deg{P}\geq
2$. Let $\mathbb{P}_{P}$ be the set of prime numbers $p\in\mathbb{P}$ such
that $P(\mathbb{Z})$ represents every residue mod $p$. That is,

$\mathbb{P}_{P}=\left\\{p\in\mathbb{P}:\forall r\in\mathbb{Z},\exists
n\in\mathbb{Z},P(n)\equiv r\pmod{p}\right\\}.$

Then

$\sum_{p\in\mathbb{P}\setminus\mathbb{P}_{P}}\frac{1}{p}=\infty.$

###### Proof.

Let us first reduce to the case that $P$ has integer coefficients. Let
$D\in\mathbb{N}$ such that $Q(x)=D\cdot P(x)\in\mathbb{Z}[x]$. For any prime
$p\in\mathbb{P}$ with $p\nmid D$, we have $p\in\mathbb{P}_{P}$ if and only if
$p\in\mathbb{P}_{Q}$, since $D$ is invertible mod $p$. Therefore,
$\mathbb{P}_{P}\subseteq\mathbb{P}_{Q}\cup\\{p\in\mathbb{P}:p\mid D\\}$. The
set $\\{p\in\mathbb{P}:p\mid D\\}$ is finite, so

$\sum_{p\in\mathbb{P}\setminus\mathbb{P}_{P}}\frac{1}{p}=\infty\iff\sum_{p\in\mathbb{P}\setminus\mathbb{P}_{Q}}\frac{1}{p}=\infty.$

We may therefore assume without loss of generality that $P$ has integer
coefficients. In this case, $P$ is well-defined mod $p$ for every
$p\in\mathbb{P}$, so $p\in\mathbb{P}_{P}$ if and only if $P$ acts as a
permutation on $\mathbb{Z}/p\mathbb{Z}$; that is, $P$ is a permutation
polynomial mod $p$. If $\mathbb{P}_{P}$ is finite, there is nothing to prove,
so suppose $\mathbb{P}_{P}$ is infinite. Then $P$ is a composition of Dickson
polynomials666The Dickson polynomials are defined recursively by
$D_{0}(a,x)=2$, $D_{1}(a,x)=x$, and $D_{n}(a,x)=xD_{n-1}(a,x)-aD_{n-2}(a,x)$,
and a Dickson polynomial $D_{n}(a,x)$ (considered as a polynomial in $x$ with
$a\in\mathbb{Z}$ fixed) is a permutation polynomial mod $p$ if and only if
either $p\mid a$ and $\gcd(n,p-1)=1$ or $p\nmid a$ and $\gcd(n,p^{2}-1)=1$;
see [14, Lemma 1.4]. and linear polynomials by [14, Theorem 2]. Hence, we may
assume $P=P_{1}\circ P_{2}\circ\dots\circ P_{r}$, where each $P_{i}(x)$ is
either linear or a Dickson polynomial. Since $\deg{P}\geq 2$, at least one of
the polynomials $P_{i}$ has $\deg{P_{i}}\geq 2$ and is therefore a Dickson
polynomial. Let $i\in\\{1,\dots,r\\}$ such that $P_{i}$ is a Dickson
polynomial, say $P_{i}=D_{n_{i}}(a_{i},x)$ for some $a_{i}\in\mathbb{Z}$ and
$n_{i}\geq 2$. If $p\in\mathbb{P}_{P}$, then $P_{i}$ must be a permutation
polynomial mod $p$, so either $p\mid a_{i}$ and $\gcd(n_{i},p-1)=1$ or $p\nmid
a_{i}$ and $\gcd(n_{i},p^{2}-1)=1$. In particular, if $q$ is a prime factor of
$n_{i}$ and $p\in\mathbb{P}_{P}$, then $q\nmid p-1$. Therefore, for any prime
factor $q\in\mathbb{P}$ of $n_{i}$,

$\\{p\in\mathbb{P}:p\equiv
1\pmod{q}\\}\subseteq\mathbb{P}\setminus\mathbb{P}_{P}.$

By Dirichlet’s theorem on primes in arithmetic progressions, it follows that

$\sum_{p\in\mathbb{P}\setminus\mathbb{P}_{P}}\frac{1}{p}=\infty.$

∎

###### Proof of Theorem 1.5.

For each $k\in\mathbb{N}$, let
$D_{k}=\bigcup_{n\in\mathbb{N}}\left(k\mathbb{N}+P(n)+n\right)$, and let
$\mathscr{D}=\left\\{D_{k}:k\in\mathbb{N}\right\\}$. Note that $D_{k}$ has
uniform density equal to $\frac{|R_{k}|}{k}$, where
$R_{k}=\left\\{P(n)+n\mod{k}:n\in\mathbb{Z}\right\\}\subseteq\mathbb{Z}/k\mathbb{Z}$.
Therefore, by Theorem 2.2 and Lemma 4.2, it suffices to show
$\inf_{k\in\mathbb{N}}\frac{|R_{k}|}{k}=0$.

Let $\mathbb{P}_{P+x}$ be the set as in Lemma 4.3 for the polynomial $P(x)+x$.
We will consider $k$ of the form $k=\prod_{i=1}^{s}p_{i}$ with
$p_{1},\dots,p_{s}\in\mathbb{P}\setminus\mathbb{P}_{P+x}$ distinct. By the
Chinese remainder theorem, $P(n)+n\equiv r\pmod{k}$ if and only if
$P(n)+n\equiv r\pmod{p_{i}}$ for every $i\in\\{1,\dots,s\\}$. Therefore,
$|R_{k}|\leq\prod_{i=1}^{s}|R_{p_{i}}|$. By the definition of
$\mathbb{P}_{P+x}$, we have $|R_{p_{i}}|\leq p_{i}-1$, so

$\frac{|R_{k}|}{k}\leq\prod_{i=1}^{s}\frac{|R_{p_{i}}|}{p_{i}}\leq\prod_{i=1}^{s}\left(1-\frac{1}{p_{i}}\right).$

We have thus shown

$\inf_{k\in\mathbb{N}}\frac{|R_{k}|}{k}\leq\prod_{p\in\mathbb{P}\setminus\mathbb{P}_{P+x}}\left(1-\frac{1}{p}\right).$

But by Lemma 4.3, the infinite product
$\prod_{p\in\mathbb{P}\setminus\mathbb{P}_{P+x}}\left(1-\frac{1}{p}\right)$ is
equal to 0, so we are done. ∎

######  Remark 4.4.

The family $\\{P(x)\in\mathbb{Q}[x]:P~{}\text{is integer-valued
and}~{}\deg{P}\geq 2\\}$ is countable, so the set $A$ in Theorem 1.5 can be
chosen independent of the polynomial $P$ by another application of Lemma 3.1.

###  4.4. Product-sum sets

######  Lemma 4.5.

Let $C\in\mathscr{C}_{\times,+}$. Then for any $k\in\mathbb{N}$, there exists
$n\in\mathbb{N}$ such that
$A\cap(k\mathbb{N}+n^{2}+n)\in\mathscr{C}_{\times,+}$.

###### Proof.

Let $B\subseteq\mathbb{N}$ be an infinite set such that
$B\footnotesize~{}\rotatebox[origin={c}]{90.0}{$\trianglecdot$}~{}B\footnotesize~{}\rotatebox[origin={c}]{90.0}{$\triangleplus$}~{}B\subseteq
A$. By the pigeonhole principle, let $n\in\\{0,1,\dots,k-1\\}$ such that
$B^{\prime}=B\cap(k\mathbb{N}+n)$ is infinite. Then
$B^{\prime}\footnotesize~{}\rotatebox[origin={c}]{90.0}{$\trianglecdot$}~{}B^{\prime}\footnotesize~{}\rotatebox[origin={c}]{90.0}{$\triangleplus$}~{}B^{\prime}\subseteq
A\cap(k\mathbb{N}+n^{2}+n)$. ∎

The family
$\mathscr{D}=\left\\{\bigcup_{n\in\mathbb{N}}\left(k\mathbb{N}+n^{2}+n\right):k\in\mathbb{N}\right\\}$
is the same family of sets that appeared in the proof of Theorem 1.5 for
$P(x)=x^{2}$, so the same argument proves Theorem 1.7.

##  5\. Concluding remarks

Each of Theorems 1.3, 1.4, 1.5, and 1.7 can be strengthened as follows.
Instead of simply concluding that $B$ is a finite set, we may make the
stronger conclusion that $|B|\leq M_{t}$, where $M_{t}$ is some finite bound
depending on $t$. That is, each shift of the set $A$ not only does not contain
any infinite configuration of the desired type, but it in fact only contains
configurations of bounded size. This can be deduced using the following
variant of Theorem 2.2:

######  Theorem 5.1.

Let $\Gamma$ be an abelian group. For each $m\in\mathbb{N}$, let
$\mathscr{C}_{m}$ be a family of subsets of $\Gamma$ and of cardinality at
least $m$ such that $\mathscr{C}_{1}\supseteq\mathscr{C}_{2}\supseteq\dots$.
Suppose there are families
$\mathscr{D}_{1}\subseteq\mathscr{D}_{2}\subseteq\dots$ of subsets of $\Gamma$
and a Følner sequence $\Phi$ with the following properties:

  1. (1)

For any $m,r\in\mathbb{N}$, there exists $M=M(m,r)\in\mathbb{N}$ such that if
$C\in\mathscr{C}_{M}$ and $D\in\mathscr{D}_{r}$, then $C\cap
D\in\mathscr{C}_{m}$, and

  2. (2)

$\inf_{r\in\mathbb{N}}\inf_{D\in\mathscr{D}_{r}}\overline{d}_{\Phi}(D)=0$.

Then for any $\varepsilon>0$, there exists $A\subseteq\Gamma$ with
$\underline{d}_{\Phi}(A)>1-\varepsilon$ and $(M_{t})_{t\in\Gamma}$ in
$\mathbb{N}$ such that for any $t\in\Gamma$, $A-t\notin\mathscr{C}_{M_{t}}$.

Theorem 5.1 can be proved along exactly the same lines as Theorem 2.2, and the
claimed strengthenings of Theorems 1.3, 1.4, 1.5, and 1.7 hold by similar
arguments to those presented in Section 4. To avoid significant repetition, we
omit the proofs and leave the details to the interested reader.

In the proof of Theorem 1.5, the reason we work with polynomials
$P(x)\in\mathbb{Q}[x]$ with $\deg{P}\geq 2$ is that the set of values
$P(\mathbb{Z})\subseteq\mathbb{Z}$ is sparse (of zero upper Banach density).
This is a key aspect of what allows for property (2) in Theorem 2.2 to hold. A
natural question, then, is whether the polynomial structure is really needed,
or if the same result holds for arbitrary sparse sequences.

######  Question 5.2.

Does there exist an increasing function $f:\mathbb{N}\to\mathbb{N}$ such that
$\lim_{n\to\infty}\frac{f(n)}{n}=\infty$ with the following property: for any
$A\subseteq\mathbb{N}$ with $d(A)>0$, there exists an infinite set
$B\subseteq\mathbb{N}$ and $t\in\mathbb{Z}$ such that

$f(B)\footnotesize~{}\rotatebox[origin={c}]{90.0}{$\triangleplus$}~{}B+t=\left\\{f(b_{1})+b_{2}+t:b_{1},b_{2}\in
B,b_{1}<b_{2}\right\\}\subseteq A?$

## Acknowledgements

This work is supported by the Swiss National Science Foundation grant
TMSGI2-211214. Thanks to Florian K. Richter and Dimitrios Charamaras for
insightful conversations about sumsets in abelian groups and for comments on
an earlier draft of the paper.

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