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Abstract
For
A
⊆
ℝ ,
let
A
+
A
=
{ a
+
b
:
a , b
∈
A }
and
A A
=
{ a b
:
a , b
∈
A } .
For k
∈
ℕ , let
SP ( k ) denote the
minimum value of
max { | A
+
A | , | A A | }
over all
A
⊆
ℕ
with | A |
=
k . Here
we establish
SP ( k )
= 3 k
− 3
for
2
≤
k
≤ 7 , the
k
= 7 case achieved
for example by
{ 1 , 2 , 3 , 4 , 6 , 8 , 1 2 } ,
while
SP ( k )
= 3 k
− 2 for
k
= 8 , 9 , the
k
= 9 case achieved
for example by
{ 1 , 2 , 3 , 4 , 6 , 8 , 9 , 1 2 , 1 6 } .
For 4
≤
k
≤ 7 ,
we provide two proofs using different applications of Freiman’s
3 k − 4
theorem; one of the proofs includes extensive case analysis on the product sets of
k -element subsets of
( 2 k − 3 ) -term arithmetic
progressions. For
k
= 8 , 9 ,
we apply Freiman’s
3 k − 3
theorem for product sets, and investigate the sumset of the
union of two geometric progressions with the same common ratio
r
> 1 ,
with separate treatments of the overlapping cases
r ≠ 2 and
r
≥ 2 .
Keywords
sum-product problem, arithmetic combinatorics
Mathematical Subject Classification
Primary: 11B30
Milestones
Received: 28 July 2023
Revised: 12 August 2023
Accepted: 18 August 2023
Published: 24 January 2025
Communicated by Nathan Kaplan
© 2025 MSP (Mathematical Sciences
Publishers).