Abstract
AbstractLet A be a subset of the cyclic group $${\textbf{Z}}/p{\textbf{Z}}$$
Z
/
p
Z
with p prime. It is a well-studied problem to determine how small |A| can be if there is no unique sum in $$A+A$$
A
+
A
, meaning that for every two elements $$a_1,a_2\in A$$
a
1
,
a
2
∈
A
, there exist $$a_1',a_2'\in A$$
a
1
′
,
a
2
′
∈
A
such that $$a_1+a_2=a_1'+a_2'$$
a
1
+
a
2
=
a
1
′
+
a
2
′
and $$\{a_1,a_2\}\ne \{a_1',a_2'\}$$
{
a
1
,
a
2
}
≠
{
a
1
′
,
a
2
′
}
. Let m(p) be the size of a smallest subset of $${\textbf{Z}}/p{\textbf{Z}}$$
Z
/
p
Z
with no unique sum. The previous best known bounds are $$\log p \ll m(p)\ll \sqrt{p}$$
log
p
≪
m
(
p
)
≪
p
. In this paper we improve both the upper and lower bounds to $$\omega (p)\log p \leqslant m(p)\ll (\log p)^2$$
ω
(
p
)
log
p
⩽
m
(
p
)
≪
(
log
p
)
2
for some function $$\omega (p)$$
ω
(
p
)
which tends to infinity as $$p\rightarrow \infty $$
p
→
∞
. In particular, this shows that for any $$B\subset {\textbf{Z}}/p{\textbf{Z}}$$
B
⊂
Z
/
p
Z
of size $$|B|<\omega (p)\log p$$
|
B
|
<
ω
(
p
)
log
p
, its sumset $$B+B$$
B
+
B
contains a unique sum. We also obtain corresponding bounds on the size of the smallest subset of a general Abelian group having no unique sum.
Publisher
Springer Science and Business Media LLC
Subject
Computational Mathematics,Discrete Mathematics and Combinatorics