On a problem of partitions of $\mathbb{Z}_{m}$ with the same representation functions

Abstract: For any positive integer

m

, let

m a t h b b Z_{m}

be the set of residue classes modulo

m

. For

A s u b s e t e q m a t h b b Z_{m}

and

o v e r l i n e n i n m a t h b b Z_{m}

, let representation function

R_{A} (o v e r l i n e n)

denote the number of solutions of the equation

o v e r l i n e n = o v e r l i n e a + o v e r l i n e a^{'}

with ordered pairs

(o v e r l i n e a, o v e r l i n e a^{'}) i n A i m e s A

. In this paper, we determine all sets

A, B s u b s e t e q m a t h b b Z_{m}

with

A c u p B = m a t h b b Z_{m}

and

| A c a p B | = 2

or

m - 2

such that

R_{A} (o v e r l i n e n) = R_{B} (o v e r l i n e n)

for all

o v e r l i n e n i n m a t h b b Z_{m}

. We also prove that if

m

is a positive integer with

4 | m

, then there exist two distinct sets

A, B s u b s e t e q m a t h b b Z_{m}

with

A c u p B = m a t h b b Z_{m}

and

| A c a p B | = 4

or

m - 4

,

B e q A + o v e r l i n e f r a c m 2

such that

R_{A} (o v e r l i n e n) = R_{B} (o v e r l i n e n)

for all

o v e r l i n e n i n m a t h b b Z_{m}

. If

m

is a positive integer with

2 | m

,

A c u p B = m a t h b b Z_{m}

and

| A c a p B | = 4

or

m - 4

, then

R_{A} (o v e r l i n e n) = R_{B} (o v e r l i n e n)

for all

o v e r l i n e n i n m a t h b b Z_{m}

if and only if

B = A + o v e r l i n e f r a c m 2

.

This page was built for publication: On a problem of partitions of $\mathbb{Z}_{m}$ with the same representation functions