Polynomial growth of sumsets in abelian semigroups

M

ELVYN

B. N

ATHANSON

I

MRE

Z. R

UZSA

Polynomial growth of sumsets in abelian semigroups

Journal de Théorie des Nombres de Bordeaux, tome

14, n

o

_{2 (2002),}

p. 553-560

<http://www.numdam.org/item?id=JTNB_2002__14_2_553_0>

© Université Bordeaux 1, 2002, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Polynomial growth

of

sumsets

in

abelian

_semigroups

par MELVYN

B. NATHANSON* et IMRE Z. RUZSA**

To Michel France

RÉSUMÉ. Soit S un

_semi-groupe

abélien et A un sous-ensemble fini de S. On

_désigne

par hA l’ensemble de toutes les sommes de h éléments de

_A,

et

_par

_|hA|

son cardinal. On montre, par des arguments élémentaires de _comptage de _points dans les

_réseaux,

qu’il

existe un

_polynôme

_p(t)

tel que pour tout entier h assez

grand

|hA|= p(h).

Plus

_{généralement,}

on étend ce résultat aux ensembles

_h1A1

+...

_{+ hr Ar}

en obtenant la croissance

polynomiale

du cardinal en termes des variables

_{h1, h2, ... , hr.}

ABSTRACT. Let S be an abelian

_semigroup,

and A a finite

sub-set of S. The sumset hA consists of all sums of h elements of

A,

with

repetitions

allowed. Let

_|hA|

denote the

_cardinality

of hA.

Elementary

lattice

_point

_arguments are used to _prove that an

ar-bitrary

abelian

_semigroup

has

_polynomial

_growth,

that _is, there

exists a

polynomial

p(t)

such that

|hA|

=

p(h)

for all

sufficiently

large

h. Lattice

_{point counting}

is also used _{to prove} that

sum-sets of the form _{h1 A1} + ... + hr Ar have multivariate

_polynomial

growth.

1. Introduction

Let

_No

denote the set of

_{nonnegative integers,}

and

_No

the set of all

k-tuples

of

_{nonnegative integers.}

_{Geometrically,}

_No

is the set of lattice

_points

in the euclidean _space

R~

that lie in the

_nonnegative

octant.

If A is a

finite,

_nonempty

subset of

_No,

then the sumset hA is the set

of all

_integers

that can be

_represented

as the sum of h elements of

_A,

with

repetitions

allowed. A classical

_problem

in additive number

_theory

concerns

the

_growth

of a finite set of

_{nonnegative integers.}

For h

_sufficiently

_large,

Manuscrit regu le 19 janvier 2001.

This work was _begun while the authors were _participants in the Combinatorial Number

The-ory Workshop, which was _organized and supported by the Erd6s Center in _{Budapest, Hungary.}

* Supported in _{part by} the Hungarian National Foundation for Scientific _Research, Grants

No. T 025617 and T 29759.

* * Supported in _{part by grants} from the PSC-CUNY Research Award _Program and the NSA Mathematical Sciences _Program.

If

_{A 1, ... , Ar}

are

finite,

_nonempty

subsets of

No

and if

hr

are

positive

integers,

then

_h,Al

+ ... + is the sumset

_consisting

of all

integers

of the form

_bl

+ ..- +

_by,

where

_by

E

_hjaj

_{for j =}

1, ... ,

r. For

hl, ... , hr

sufficiently

large,

the structure of this "linear form" has also been

completely

determined

_(Han,

_Kirfel,

and Nathanson

_[2]),

and its

_cardinality

is a linear function of

h 1, ... ,

he.

If A is a

finite,

_nonempty

subset of

No,

the

geometrical

structure of the

sumset hA is

_complicated,

but the

_cardinality

of hA is a

polynomial

in h

of

_degree

at most 1~ for h

_{sufficiently large}

_(Khovanskii

_[3]).

If the set A is

not contained in a

_hyperplane

of dimension k -

_1,

then the

_degree

of this

polynomial

is

_{exactly equal}

to k.

The sets

_No

and

_No

are abelian

semigroups,

that

is,

sets with a

_binary

operation,

called

_addition,

that is associative and commutative. Let S be

an

arbitrary

abelian

semigroup.

Without loss of

generality,

we can assume

that S contains an additive

_identity

0. If A is a

finite,

_nonempty

subset

of S and h a

_{positive integer,}

we

again

define the sumset hA as the set

of all sums of h elements of

A,

with

_repetitions

allowed. Khovanskii

[3,

4]

made the remarkable observation that the

_cardinality

of hA is a

_polynomial

in h for all

_sufficiently

_large

_h,

that

_is,

there exists a

_polynomial

p(t)

and an

_integer

ho

such that

IhAI =

p(h)

for h &#x3E;

ho.

Khovanskii

proved

this

result

_{by constructing}

a

_{finitely generated graded}

module M =

Mh

over the

polynomial ring

C[ti , ... , tk],

7 where

JAI =

k,

with the

property

that the

_homogeneous

_component

_Mh

is a vector _space over C of dimension

exactly

IhAI

for all h &#x3E; 1. A theorem of Hilbert asserts that

_{dimc Mh}

is a

polynomial

in h for all

_{sufficiently large h,}

and this

_gives

the result.

If

_{A,7 ...}

_Ar

are

finite,

_nonempty

subsets of an abelian

_{semigroup ,S,}

and

if

_{hi , ... , hr}

are

_{positive integers,}

then the "linear form"

_h,Al

+ ... +

is the sumset

_consisting

of all elements of S of the form

_bl

+... +

_6r?

where

by

E

_{for i =}

_{1, ... ,}

r.

Using

a

generalization

of Hilbert’s theorem

to

_finitely

_generated

modules

_graded

_by

the

_semigroup

_No,

Nathanson

_[6]

proved

that there exists a

polynomial

p(t, 7.

to)

such that -~- ~ ~ ~ +

=

p ( h 1, ... ,

for all

sufficiently

large

integers h 1, ... ,

he.

The _purpose of this note is to

_{give elementary}

combinatorial

_proofs

of the theorems of Khovanskii and Nathanson that avoid the use of Hilbert

polynomials.

Our

_arguments

reduce to an _easy

_computation

about lattice

points

in euclidean _space.

2. Growth of sumsets

We

_begin

with some

_geometrical

lemmas about lattice

_points.

Let x =

(x 1, ... ,

and y

=

( y1, ... ,

be elements of

The set

_Q(h)

is a finite set of lattice

points

whose

_cardinality

is the number of ordered

partitions

of h as a sum of k

nonnegative

integers,

and so

which is a

polynomial

in h for fixed k.

We define a

partial

order on

Nk

by

In

_N§,

for

_example,

_{(2, 5) (4, 6)}

and

_{(4, 3)}

_{(4, 6),}

but the lattice

_points

(2, 5)

and

_{(4, 3)}

are

incomparable. Thus,

the

relation z

_y is a

_partial

order but not a total order. We write x _y if x _{y y. y,}

Lemma 1. Let W

_finite

subset

_of

_No,

and let

_B(h,W)

be the set

_of

all lattice E

_a(h)

such

_for

all w E W. Then

_[

is ca

polynomial in h for

all

sufficiently large

h.

and so

for h &#x3E;

_{ht(w* ) .}

This

_completes

the

_proof.

0

An ideal in an abelian

semigroup

is a

_nonempty

set I such that if x E

_I,

then x+t E I for _every element t in the

_semigroup.

In the

_partially

ordered

semigroup

Nk,

a

_nonempty

set I is an ideal if and

_only

if x E I and

_{y &#x3E;}

x

imply y

E I. The

_following

result about lattice

_points

and

_partial

orders is known as Dickson’s lemma

_[1].

We include a

proof

for

completeness.

Proof.

The

_proof

is

_by

induction on the dimension k. If k =

_1,

then I is _a

nonempty

set of

_{nonnegative integers,}

hence contains a least

integer

w. If

x &#x3E; _w, then x E I since I is an

_ideal,

and so I =

~x

_E

w~.

Let k &#x3E;

_2,

and assume that the result holds for dimension k -1. We shall

write the lattice

_point

x =

(xl, ... ,

_Xk-1)

Xk)

_E

Nk in

₀ the form x

= (x’, xk),

where x’ =

_{(x1, ... ,}

_No-1.

Define the

_projection

_map 7r :

by

1r(x)

=

x’.

Let I’ =

x(I)

be the

image

of the ideal

I,

that

is,

We have

_{I’ ~ ~}

since

_{I ~ 0.}

Let x’ E I’

and y’

E

No-1.

Since x’

E

_I’,

there is a

_{nonnegative integer}

_x~ such that E I.

_{If y’ &#x3E;}

_x’,

then

y’, xk &#x3E; x’, xk

in

_N§,

and so

(y’, Xk) E

_I,

hence y’

E I’.

_Thus,

I’ is

an ideal in Since the Lemma holds in dimension k -

1,

there is a finite set W’ C I’ such that x’

E I’

if and

only

if x’ &#x3E; w’ for some

w’ E W’. Associated to each lattice

_point

w’ E W’ is a

nonnegative integer

such that

_{(w’, xk (w’) )}

E I. Let m = w’ _E

W’}

and

Wm

_{= ~ (w’, m) : w’}

E

_W’}.

If w’ E

_W’,

then

_{(w’, m) &#x3E; (w’, zk (w’))}

and so

(w’, m)

E I.

_Therefore,

_Wm

C I.

For 1 =

0,1,... ,

_{m -}

1,

we consider the set

If

_Ie

=

0,

let

0.

0,

then

I~

is an ideal in

No-1,

and there is a

finite set

_W§

such that x’ E

_Ie

if and

_only

if x’ &#x3E; w’ for some w’ e

_Wi.

Let

W~ _ ~ (w’, ~) :

w’ E Then

_Wi

C I. We consider the set

which is a finite subset of the ideal I.

We shall _prove that x E I if and

_only

if x &#x3E; w for some w E W*. If

X =

(X’, Xk) E I

and _m, then

x’

_E

I’,

hence x’ &#x3E;

w’

for some 2v’ E W’.

It follows that

and

_{(w’, m)}

E

_Wm

9

W *.

and so x’ &#x3E;

w’

for some

If x =

(x’,E)

_E I and 0 .~ _m, then x’ _E

Ie,

and so

x’ &#x3E; w’

for some

w’ E

_{W~ .}

It follows that

Let x =

(x 1, ... ,

Xk)

and _y =

( yl, ... ,

be lattice

points

in

N§.

We

define the

_{lexicographical}

order on

N§

as follows: if either X =

y or there

k~

such

that Xi

= _Yi for i =

1, ... , j -1

and

Xj Yj. This is a total

order,

so _every

finite,

_nonempty

set of lattice

_points

contains a smallest lattice

point.

For

example,

(2,5)~r(4,3)~(4,6).

If

then X + + t for all t E

_N§.

We write if and Theorem 1. Let S be an abelian

semigroup,

and let A be

a finite

nonempty

subset

_of

S. There exists a

polynomial

p(t)

such that

p(h)

f or

all

sufficiently large

h.

Proof.

Let A =

~al, ... , ak ~,

where

IAI

= k. We define _{a map}

f :

N§

_-~ S

as follows: If X =

(x 1, ... ,

_E

N§,

then

This is

_{well-defined,}

since each xi is a

_{nonnegative integer}

and we can add

the

_semigroup

_{element ai}

to itself ~i times. The map

f

is a

_homomorphism

of

_semigroups:

If _x, y E

No,

then

_{f (x + y)}

= ₊

f (y).

We consider the set

If x E

_Q(h),

then

_{f (x)}

E hA and = hA. The _map

f

is not

necessarily

one-to-one on the set

_Q(h).

For any s E

_hA,

there can be _many

lattice

_points

E

_Q(h)

such that = s.

_However,

for each s E

_hA,

there

is a

_unique

lattice

point

uh(s)

E

f-l(8)

_na(h)

that is

_{lexicographically}

smallest,

that

_is,

for all x

_{na (h).}

Then

The lattice

_{point x}

E

Nk

will be called useless

_if,

for h =

ht(x),

we have

x =1=

Uh (s)

for all s E hA.

_{Equivalently, x}

E

Nk

is useless if there exists a

lattice

_point

u E such that

_f(u)

=

f (x)

and Let I be the set of all useless lattice

_points

in

_No.

We shall prove that I is an ideal in the

_semigroup

No.

Let x E

_I,

ht(x)

=

h,

and t _E

Nk.

Since _{x E}

I,

there exists a lattice

_point

u E

_Q(h)

such that

_{f (u)}

=

f (x)

and Then

It follows that x + t is

useless,

hence x + t E I and I is an ideal of the

semigroup

No.

We call I the useless ideal.

By

Dickson’s lemma

_{(Lemma 2),}

there is a finite set W* of lattice

_points

in

_No

such that x E

N kis

useless if and

_only

if x &#x3E; zu for some w E W*. The

_cardinality

of the sumset hA is the number of lattice

_points

in

_a(h)

that are not in the useless ideal I. For _every subset W C

_W*,

we define

the set

By

the

_principle

of

_{inclusion-exclusion,}

By

Lemma

_1,

for _every W C W * there is an

_integer

ho (W )

such that

I B (h, W)

I

is a

polynomial

in h for h &#x3E;

ho(W).

_Therefore,

IhAI

I

is a

poly-nomial in h for all

_{sufficiently large}

h. This

_completes

the

_proof.

D

3. Growth of linear forms

Let

_{kl, ... ,}

_k,.

be

_{positive integers,}

and let k =

kl

_{+ ... +}

k,..

We shall

write the

_semigroup

_Nkin

the form

and denote the lattice

_point

x E

No

_{by x}

=

(xl .... xr),

where xj _E

No’

for j =

1, ... ,

r.

Let hj =

ht (z; )

for j =

_{l, ... ,}

r. We define the

r-height

of x

_by

_{htr(x) =}

_{(hl, ... , hr).}

For _any

_positive

_integers

_{hl, ... , hr,}

we

consider the set

Then

is a

polynomial

in the r variables

_{hl, ... , hr}

for fixed

_{integers ki , ... ,}

_kr.

Lemma 3. Let

_{1~1, ... , kr}

be

_positive

_integers,

and k =

kl

_{+ ..- +}

kr.

Let

W be a

finite subset of

No

=

Nol

x - - - x and let

_{B(h1, ... ,}

_hr,

_{W )}

be the set

_of

all lattice

_{points x}

E

No

such that x E

_{a( hI, ... , hr)}

and

)jB(i,...

a

_polynomial

in

_{hi , ... , hr}

_,for

all

_sujficiently

_large

integers

h1, ... , h~..

Proof.

Let x =

(xI , ... ,

_E Let

W;

be the set of all lattice

_points

Wj E

N§~

such that there exists a lattice

_point

w E W of the form w =

( w 1, ... ,

w~ , ... ,

wr ) .

for all w E W if and

_only

if for all _Wj E

_W;,

it follows that the set

_{2?(~i,... ,}

_hr,

_{W )}

consists of all lattice

points

x =

(x1, ... ,

Xr) E

N©

such

that X j

_E

B (hj,

for

all j =

1, ... ,

r.

Therefore,

It follows from Lemma 1 that

_{hr, W)I}

is a

_polynomial

in the

r variables

hl, ... , hT

for all

sufhciently large integers

hl, ... , hr.

This

completes

the

_proof.

0

Theorem 2. Let S be an abelian

semigroup,

and let

~4i,...

Ar

be

fi-nite, nonempty

subsets

_of

S. There exists a

_polynomial

p(tl,

_{... ,}

tr)

such

that

_lh,Al

+ ... +

_p(hi,

... ,

hr)

for

all

sufficiently large

integers

hl,... hr .

Let k =

k1

_{+ ... +}

k,..

We consider lattice

points

where

Define the

_{semigroup homomorphism f :}

S’ as follows: If x =

(xl, ... , Xr)

E No,

then

A lattice

_{point x}

E

N©

will be called r-useless if there exists a lattice

point

u E

_a(htr(x))

such that

_{f(u) =}

and As in the

_proof

of Theorem

_1,

the set

_Ir

of useless lattice

_points

in

_N§

is an ideal.

_By

Lemma

_2,

there is a finite set W * that

_{generates Ir}

in the sense that x E

N§

is r-useless if and

_only

if

_{x &#x3E;}

w for some w e W * . Let

_{( h I , ... ,}

_{E No}

and

subset W C

_{W * ,}

we define the set

By

the

_principle

of

_{inclusion-exclusion,}

By

Lemma

_3,

for all

_{sufficiently large integers hl, ... , he,}

the function

[B (hi , ... ,

is a

polynomial

in

hl, ... ,

h,.,

and so

I

is a

polynomial

in

hl, ... ,

he.

This

completes

the

proof.

0

Remark. It would be

_interesting

to describe the set of

_polynomials

_{f (t)}

such that

_{f (h) =}

_(M)

for some finite set A and

_{sufficiently large}

h.

_Similarly,

one can ask for a

description

of the set of

_polynomials

_{f (tl, ... ,}

such that

_{f (h1, ... ,}

_{hr) _}

where

_{~4i,... ,}

_Ar

are finite subsets

of a

_semigroup

S.

References

[1] D. Cox, J. _LITTLE, D. _{O’SHEA, Ideals, Varieties,} and _{Algorithms. Springer-Verlag,} New

York, 2nd _edition, 1997.

[2] S. HAN, C. _KIRFEL, M. B. _NATHANSON, Linear _forms in finite sets _{of integers. Ramanujan}

J. 2 _(1998), 271-281.

[3] A. G. KHOVANSKII, Newton polyhedron, Hilbert polynomial, and sums _{of finite} sets. Func-tional. Anal. Appl. 26 _(1992), 276-281.

[4] A. G. KHOVANSKII, Sums _{of finite} sets, orbits _of commutative semigroups, and Hilbert func-tions. Functional. Anal. _Appl. 29 _(1995), 102-112.

[5] M. B. NATHANSON, Sums of finite sets _{of integers.} Amer. Math. Monthly 79 _(1972),

1010-1012.

[6] M. B. NATHANSON, Growth of sumsets in abelian semigroups. Semigroup Forum 61 _(2000),

149-153.

Melvyn B. NATHANSON

Department of Mathematics Lehman _{College (CUNY)}

Bronx, New York 10468

USA

E-mail : _{nathansnealpha -} lehman . cuny . edu

Imre Z. RuzsA

Alfr6d R6nyi Mathematical Institute

Hungarian Academy of Sciences

Budapest, Pf. 127 H-1364 _Hungary E-mail : _{ruzsa0renyi . hu}