Abstract
We study the structure of the difference set
E
−
E
$E - E$
upper E minus upper E
for subsets
E
⊆
Z
2
$E \subseteq \mathbb {Z}^2$
upper E subset of or equal to double struck upper Z squared
of positive upper Banach density. Fish [‘On product of difference sets for sets of positive density’,
Proc. Amer. Math. Soc.
146
(2018), 3449–3453] asked in Problem 2 whether, for every such set
E
, there exists a nonzero integer
k
such that
k
⋅
Z
⊆
{
x
y
:
(
x
,
y
)
∈
E
−
E
}
$k \cdot \mathbb {Z} \subseteq \{ xy : (x,y) \in E - E \}$
k dot double struck upper Z subset of or equal to StartSet x y colon left parenthesis x comma y right parenthesis element of upper E minus upper E EndSet
. Although this question remains open, we establish a relatively weaker form. Specifically, we prove that if
⟨
a
j
⟩
j
=
1
m
$\langle a_j\rangle _{j=1}^m$
left angle bracket a Subscript j Baseline right angle bracket Subscript j equals 1 Superscript m
is any finite sequence in
N
,
$\mathbb {N},$
double struck upper N comma
then there exist infinitely many integers
k
∈
Z
$k \in \mathbb {Z}$
k element of double struck upper Z
and infinitely many sequences
⟨
x
n
⟩
n
∈
N
$\langle x_n \rangle _{n \in \mathbb {N}}$
left angle bracket x Subscript n Baseline right angle bracket Subscript n element of double struck upper N
in
Z
$\mathbb {Z}$
double struck upper Z
such that
k
⋅
M
T
(
⟨
a
j
⟩
j
=
1
m
,
⟨
x
n
⟩
n
)
⊆
{
x
y
:
(
x
,
y
)
∈
E
−
E
}
$k \cdot MT(\langle a_j \rangle _{j=1}^m, \langle x_n\rangle _{n}) \subseteq \{ xy : (x,y) \in E - E \}$
k dot upper M upper T left parenthesis left angle bracket a Subscript j Baseline right angle bracket Subscript j equals 1 Superscript m Baseline comma left angle bracket x Subscript n Baseline right angle bracket Subscript n Baseline right parenthesis subset of or equal to StartSet x y colon left parenthesis x comma y right parenthesis element of upper E minus upper E EndSet
, where
M
T
(
⟨
a
j
⟩
j
=
1
m
,
⟨
x
n
⟩
n
)
$MT(\langle a_j \rangle _{j=1}^m, \langle x_n\rangle _{n})$
upper M upper T left parenthesis left angle bracket a Subscript j Baseline right angle bracket Subscript j equals 1 Superscript m Baseline comma left angle bracket x Subscript n Baseline right angle bracket Subscript n Baseline right parenthesis
denotes the Milliken–Taylor configuration generated by the sequences
⟨
a
j
⟩
j
=
1
m
$\langle a_j\rangle _{j=1}^m$
left angle bracket a Subscript j Baseline right angle bracket Subscript j equals 1 Superscript m
and
⟨
x
n
⟩
n
∈
N
$\langle x_n \rangle _{n \in \mathbb {N}}$
left angle bracket x Subscript n Baseline right angle bracket Subscript n element of double struck upper N
.