Long arithmetic progressions in sumsets: Thresholds and Bounds
Abstract
For a set
A
of integers, the sumset
lA=A+...+A
consists of those numbers which can be represented as a sum of
l
elements of
A
lA={
a
1
+...
a
l
|
a
i
∈
A
i
}.
A closely related and equally interesting notion is that of
l
∗
A
, which is the collection of numbers which can be represented as a sum of
l
different elements of
A
l
∗
A={
a
1
+...
a
l
|
a
i
∈
A
i
,
a
i
≠
a
j
}.
The goal of this paper is to investigate the structure of
lA
and
l
∗
A
, where
A
is a subset of
{1,2,...,n}
. As applications, we solve two conjectures by Erdös and Folkman, posed in sixties.