Constructing finite sets from their representation functions

• Published in: Acta mathematica Hungarica Vol. 165; no. 1; pp. 134 - 145
• Main Authors: Sun, C.-F., Xiong, M.-C., Yang, Q.-H.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-10-2021; Springer Nature B.V
• Subjects: Mathematics; Mathematics and Statistics; Representations; partition; representation function; 11B13; residue class
• ISSN: 0236-5294; 1588-2632
• DOI: 10.1007/s10474-021-01183-1

For any positive integer
m
, let
Z
m
be the set of residue classes modulo
m
. For
A
⊆
Z
m
and
n
¯
∈
Z
m
, let the representation function
R
A
(
n
¯
)
denote the number of solutions of the equation
n
¯
=
a
¯
+
a
′
¯
with unordered pairs
(
a
¯
,
a
′
¯
)
∈
A
×
A
. We characterize the partitions of
Z
2
p
with
A
∪
B
=
Z
2
p
and
|
A
∩
B
|
=
2
such that
R
A
(
n
¯
)
=
R
B
(
n
¯
)
for all
n
¯
∈
Z
2
p
, where
p
is an odd prime.