Sidon set

Sidon set

A set of natural numbers is called a

Sidon set

if all
pairwise sums of its

elements

are distinct. Equivalently, the equation

a

+

b

=

c

+

d

has only the

trivial

solution

{

a

,

b

}

=

{

c

,

d

}

in
elements of the set.

Sidon sets are a special case of so-called

B

h

⁢

[

g

]

sets. A set

A

is called a

B

h

⁢

[

g

]

set if for every

n

∈

ℕ

the equation

n

=

a

1

+

…

+

a

h

has at most

g

different solutions with

a

1

≤

⋯

≤

a

h

being elements of

A

. The Sidon sets are

B

2

⁢

[

1

]

sets.

Define

F

h

⁢

(

n

,

g

)

as the size of the largest

B

h

⁢

[

g

]

set

contained

in the interval

[

1

,

n

]

. Whereas it is known that

F

2

⁢

(

n

,

1

)

=

n

1

/

2

+

O

⁢

(

n

5

/

16

)

[

3

, p. 85]

,
no asymptotical results are known for

g

>

1

or

h

>

2

[

2

]

.

Every

finite set

has a rather large

B

h

⁢

[

1

]

subset. Komlós,
Sulyok, and
Szemerédi

[

4

]

proved
that in every

n

-element set there is a

B

h

⁢

[

1

]

subset of size at
least

c

⁢

F

h

⁢

(

n

,

1

)

for

c

=

1

8

⁢

(

2

⁢

h

)

-

6

. For Sidon sets
Abbott

[

1

]

improved that to

c

=

2

/

25

.
It is not known whether one can take

c

=

1

.

The

infinite

B

h

⁢

[

g

]

sets are understood even worse. Erdős

[

3

, p. 89]

proved that for every
infinite Sidon set

A

we have

lim inf

n

→

∞

⁡

(

n

/

log

⁡

n

)

-

1

/

2

⁢

|

A

∩

[

1

,

n

]

|

<

C

for some

constant

C

. On the other
hand, for a long time no example of a set for which

|

A

∩

[

1

,

n

]

|

>

n

1

/

3

+

ϵ

for some

ϵ

>

0

was known. Only
recently Ruzsa

[

6

]

used an extremely
clever construction to prove the

existence

of a set

A

for which

|

A

∩

[

1

,

n

]

|

>

n

2

-

1

-

ϵ

for every

ϵ

>

0

and for all sufficiently large

n

.

For an excellent survey of Sidon sets see

[

5

]

.

References

1

Harvey Leslie Abbott.

Sidon sets.

Canad. Math. Bull.

, 33(3):335–341, 1990.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0715.11004

Zbl 0715.11004.

2

Ben Green.

B

h

⁢

[

g

]

sets: The current state of affairs.

http://www.dpmms.cam.ac.uk/ bjg23/papers/bhgbounds.dvi

http://www.dpmms.cam.ac.uk/ bjg23/papers/bhgbounds.dvi, 2000.

3

Heini Halberstam and Klaus Friedrich Roth.

Sequences

.

Springer-Verlag, second edition, 1983.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0498.10001

Zbl 0498.10001.

4

János Komlós, Miklos Sulyok, and Endre Szemerédi.

Linear problems in combinatorial number theory.

Acta Math. Acad. Sci. Hung.

, 26:113–121, 1975.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0303.10058

Zbl 0303.10058.

5

Kevin O’Bryant.

A

complete

annotated bibliography of work related to Sidon
sequences.

Electronic Journal of Combinatorics

.

http://arxiv.org/abs/math.NT/0407117

http://arxiv.org/abs/math.NT/0407117.

6

Imre Ruzsa.

An infinite Sidon sequence.

J.

Number Theory

, 68(1):63–71, 1998.

Available at

http://www.math-inst.hu/ ruzsa/cikkek.html

http://www.math-inst.hu/ ruzsa/cikkek.html.

Title

Sidon set

Canonical

name

SidonSet

Date of creation

2013-03-22 13:59:40

Last modified on

2013-03-22 13:59:40

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

12

Author

bbukh (348)

Entry type

Definition

Classification

msc 11B05

Classification

msc 11B34

Synonym

Sidon sequence

Related topic

ErdHosTuranConjecture

Defines

B

h

⁢

[

g

]

set

Generated on Fri Feb 9 15:25:51 2018 by

LaTeXML