oscillations of bases for the natural numbers - American Mathematical ...

proceedings of the american mathematical

society

Volume 53, Number 2, December

1975

OSCILLATIONS OF BASES FOR THE NATURAL NUMBERS PAUL ERDOS AND MELVYN B. NATHANSON ABSTRACT. sufficiently

with

a.,

a.

sets

which

basis

€ A.

oscillate

If A = ftf-l1*,

2, or simply,

or, simply,

and also

that

is, A ij \b\

under

[lL

then infinitely

number

n can be

basis

of order

many numbers nonbasis

is a basis

subset

are

of order, 2,

Minimal

bases

for every

were

that

introduced

examples

by

of minimal

is minimaL

if no proper number

[3], and examples not known

of A is a basis;

L3J constructed

of which

is maximal

It is still

bases

small

Such

Notation.

and maximal

superset

b 4 A.

were

if every

of A is a non-

Maximal

nonbases

constructed

nonbasis

Latin

of numbers

are examples

from bases

oscillations

are the theme

Numbers

case

nonbases

perturbations

will

letters.

by \A\ the cardinality set

large

by Erdos

is contained

in a

nonbasis„

Minimal

The

to

A set is a set of numbers.

an asymptotic

a . £ A.

no subset

by Nathanson

and Nathanson

by upper

for every

A = {«■(• _j

introduced

bases.

or from nonbasis

an asymptotic

if no proper

[2] and Nathanson

of bases

A nonbasis

maximal

is minimal

is a nonbasis

L4], and Hartter

were

n = a. + a.

we construct

sets.

sufficiently

A is called

A is called

paper

integer,,

If A is not a basis,

a . + a ., and

A = l«.i°c_.j

A\|fl.S

basis;

every

A is a basis

form

a nonbasis.

A basis

bases,

that

In this

of the

is a positive

such

Then in the

to basis

perturbations

n = a. + a ., then

a basis.

not of the form

late

a set

integers.

be written

to nonbasis

finite

A number

is

n can

A is a nonbasis.

from basis under

in the form

of positive

integer

Otherwise,

Introduction.

written

Stohr

A be a set large

to nonbasis

1.

is,

Let

if every

be denoted

The

of the set

between

of this

by lower

case

set of all numbers

b is denoted

which

oscil-

and from nonbases

to

paper.

A, and by A\B

a and

of sets

to nonbases

Latin

letters,

and sets

N.

We denote

is denoted

the complement [a, b\

of B in A.

If A = \a.\oc_,

and

B = \b .\°° ,, then the sum of A and B is A + B = \a . + b .\a . e A, b. £ B\. The sum

A + A is written

do not exceed

n

2A.

is denoted

Finally,

A(n),

the number

and the set

of elements

A has

density

of A which

8 it

limn—oc Ain)/n = 8. Received

by the editors

AMS (MOS) subject 10L15. Key words oscillations

and phrases.

of bases,

October

classifications

additive

Minimal number

21, 1974.

(1970). basis,

Primary 10L05, 10J99; Secondary

maximal

nonbasis,

sumsets

of integers,

theory.

Copyrighr © 1975, American Mathematical License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

253

Society

254

PAUL ERDOS AND M.B.NATHANSON Lemma.

Let

5?fe„,+3.

0 = \2q,

+ 1 \T_,

be a set

of odd numbers

such

that

q, >

Let CO

A° Then

2A

n\Q.

U \[2qk_l+2,

C N\0,

Moreover,

2(A^\F)

and

2A

contains

if F and

G

differ from 2/1

|2(A°U G)\2iAQ\F)\

qk-qk_1]u[qk+l,

arc

qk + qk_i]\.

all but finitely

any

finite

by only finitely

many

sets,

then

of the numbers

2(A^

many numbers;

in

U G") and

that is,