Behrend’s construction

Behrend’s construction

At first sight it may seem that the greedy algorithm yields the densest subset of

{

0

,

1

,

…

,

N

}

that is free of

arithmetic progressions

of length

3

. It is not hard to show that the greedy algorithm yields the set of numbers that lack digit

2

in their ternary development (

http://planetmath.org/Base3

). Density (

http://planetmath.org/AsymptoticDensity

) of such numbers is

O

⁢

(

N

log

3

⁡

2

-

1

)

.

However, in 1946 Behrend

[

1

]

constructed much denser subsets that are free of arithmetic progressions. His major idea is that if we were looking for a progression-free sets in

ℝ

n

, then we could use spheres. So, consider an

d

-dimensional cube

[

1

,

n

]

d

∩

ℤ

d

and family of spheres

x

1

2

+

x

2

2

+

⋯

+

x

d

2

=

t

for

t

=

1

,

…

,

d

⁢

n

2

. Each point in the cube is

contained

in one of the spheres, and so at least one of the spheres contains

n

d

/

d

⁢

n

2

lattice points

. Let us call this set

A

. Since sphere does not contain arithmetic progressions,

A

does not contain any progressions either. Now let

f

be a

Freiman isomorphism

from

A

to a subset of

ℤ

defined as follows. If

x

=

{

x

1

,

x

2

,

…

,

x

d

}

is a point of

A

, then

f

⁢

(

x

)

=

x

1

+

x

2

⁢

(

2

⁢

n

)

+

x

3

⁢

(

2

⁢

n

)

2

+

⋯

+

x

d

⁢

(

2

⁢

n

)

d

-

1

, that is, we treat

x

i

as

i

’th digit of

f

⁢

(

x

)

in base

2

⁢

n

. It is not hard to see that

f

is indeed a Freiman isomorphism of order

2

, and that

f

(

A

)

⊂

{

1

,

2

,

…

,

N

=

(

2

n

)

d

}

. If we set

d

=

c

⁢

ln

⁡

N

, then we get that there is a progression-free subset of

{

1

,

2

,

…

,

N

}

of size at least

N

⁢

e

-

ln

⁡

N

(

c

ln

2

+

2

/

c

+

o

(

1

)

. To maximize this value we can set

c

=

2

/

ln

⁡

2

. Thus, there exists a progression-free set of size at least

N

⁢

e

-

8

⁢

ln

⁡

2

⁢

ln

⁡

N

⁢

(

1

+

o

⁢

(

1

)

)

.

This result was later generalized to sets not containing arithmetic progressions of length

k

by Rankin

[

3

]

. His construction is more complicated, and depends on the estimates of the number of

representations

of an

integer

as a sum of many squares. He proves that the size of a set free of

k

-term arithmetic progression is at least

N

⁢

e

-

c

⁢

(

log

⁡

N

)

1

/

(

k

-

1

)

.

On the other hand, Moser

[

2

]

gave a
construction analogous to that of Behrend, but which was explicit
since it did not use the

pigeonhole principle

.

References

1

Felix A. Behrend.

On the sets of integers which contain no three in arithmetic
progression.

Proc. Nat. Acad. Sci.

, 23:331–332, 1946.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0060.10302

Zbl 0060.10302.

2

Leo Moser.

On non-averaging sets of integers.

Canadian J. Math.

, 5:245–252, 1953.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0050.04001

Zbl
0050.04001.

3

Robert A. Rankin.

Sets of integers containing not more than a given number of terms in

arithmetical

progression.

Proc. Roy. Soc. Edinburgh Sect. A

, 65:332–344, 1962.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0104.03705

Zbl 0104.03705.

Title

Behrend’s construction

Canonical

name

BehrendsConstruction

Date of creation

2013-03-22 13:40:53

Last modified on

2013-03-22 13:40:53

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

11

Author

bbukh (348)

Entry type

Result

Classification

msc 05D10

Classification

msc 11B25

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