OSCILLATIONS OF BASES FOR THE NATURAL NUMBERS. PAUL ERDOS AND MELVYN B. NATHANSON. ABSTRACT. Let A be a set of positive integers.
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,