PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 96, Number
1. January 1986
DIVISIBILITYPROPERTIES OF ADDITIVEBASES
MELVYN B. NATHANSON
Abstract.
Let h ^ 2. There exists an asymptotic basis A of order h such that
(a¡.ak
) > 1 for all a¡.ak
s A if and only if k < ft. If k » ft, the sumset
hA contains only composite numbers. For h = k. there exists a set A of nonnegative
integers with (ax,...,ah)>
1 for all a,,..., ah = A such that for every prime p the
sumset hA contains all sufficiently large multiples of p.
1. Introduction. Let A be a set of nonnegative integers. The h-fold sum of A,
denoted hA, is the set consisting of all sums of h not necessarily distinct elements of
A. If hA contains all sufficiently large integers, then A is an asymptotic basis of order
h.
The set A has the property GCD(k) if the greatest common divisor of any k
elements of A is strictly greater than 1. In this paper it is proved that there exists an
asymptotic basis A of order h such that A satisfies GCD(k) if and only if
1 < k < h. In particular, if A satisfies GCD(k) and h < k, then no prime can be
represented as the sum of h elements of A. On the other hand, if h > 2, then there
exists a set A satisfying GCD(h) such that for every prime p there exists a number
N(p) such that np e hA for every n > N(p).
Notation. Let [1, t] denote the interval of integers 1, 2, ...,t. Let |5| denote the
cardinality of the finite set S.lî S e T, then T\ S denotes the relative complement
of S in T. If A is a set of integers, then A(x) denotes the number of positive
elements of A not exceeding x. If P is the set of primes, then tr(x) = P(x). The
upper asymptotic density of the set A is du(A) = lim sup,. _x A(x)/x. The asymptotic density of A is d(A) = lim^^
A(x)/x, if the limit exists.
This paper originated in discussions with Gerald Myerson at the Conference on
Analytic Number Theory at Oklahoma State University in June, 1984.
2. Results.
Theorem 1. Let 1 < k < h. There exists an asymptotic basis A of order h such that
(ax,a2,...,ak)>
1 for all a x, a2,...,ak
e A.
Proof. It suffices to consider only the case h = k + 1. Let qx, q2,...,qk + x be
pairwise relative prime integers greater than 1. Define sx, s2,...,sk + x by s, =
Received by the editors March 18, 1985.
1980 MathematicsSubjectClassification.Primary 10L02,10L05; Secondary 10J99,05A05.
Key words and phrases. Density of sumsets, asymptotic
bases, additive number theory, additive bases.
©
1986
American
0002-9939/86
11
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M. B. NATHANSON
12
Uk:i^,qj.
Then
(1)
(sx,...,sk
+ x) = \
and
(2)
(sx,...,sJ,...,sk
+ x)>i
for j = 1,..., k + 1, where (sx, ...,s,,...,
sk + x) > 1 denotes the greatest common
divisor of the k integers s¡ for i = 1,..., k + 1, i ¥=j.
Let A = [s¡u\u > 0,i ■»1,...,*
+ 1}. It follows from (1) that the linear form
s,m, + 52t/2 + • • • +^^ + 1"^ + ! represents all sufficiently large integers, and so A is
an asymptotic basis of order h — k + 1. Moreover, (2) implies that (ax,..., at) > 1
for all ax,...,akeA.
This proves the theorem.
Remark. The preceding proof is based on the following combinatorial fact: Let
1 « s < t and T = [1,t]. Let S¡ ç T and 15,1= s for i = 1,..., k. If k < t/(t - s),
then fifes',
# 0.
If * > t/(t - s), then there exist sets Sx,...,Sk
such that
n?_,s,= 0.
Lemma. Let h > 2. TAere ejtw/s a family ÍF of finite sets of positive integers such
that (a) the intersection of any h sets in ?F is nonempty, and (b) for any positive
integer n there exist h sets in J^ whose intersection is exactly {n}.
Proof.
Let N = Mx U • ■• U Mh be a partition of the positive integers into h
infinite sets M¡. Let M¡ = {miJ}flx. For s = l,...,h
family of finite sets Fst by
Fs,=
{mIJ\i
= \,...,h,i±s;
and t = 1,2,3,...,
define the
j=\,...,t}\j{ms,}.
Then \FS,\ = (h - l)t + 1.
Let Fst e & for ; = 1,..., h. If {sx,s2,.-.,sh}
that tj = min{ tx, t2,...,th}. Then
= [1, A], choose j e [l,h] such
ft
«m, e n ^v,# 0 •
;= 1
If (i,,..
.,sh) # [1, A], choose s* e [1, A] such that s* ¥=s¡ for ; = 1,..., A. Then
ft
1=1
Thus, the intersection of every A element of & is nonempty. This proves (a).
Let n e N. Then « = wjr for some unique s e [1, A] and ; e N, and
/
F.,n
ft
f)F,, +i
\
= ims,}
= {«}•
\'=1
^ i^i
This proves (b).
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13
DIVISIBILITY PROPERTIES OF ADDITIVE BASES
Theorem 2. Let A ^ 2. There exists a set A of nonnegative integers such that
(i) (ax,..., ah) > 1 for all a,, ...,ahe
A, and
(ii) for every prime p there is an integer N( p ) such that np e hA for all n > N(p).
Proof. Let & be a family of finite subsets of N that satisfies properties (a) and
(b) of the Lemma. For FeJ,
define the integer b(F) by b(F) = YlieFp¡, where p¡
is the /th prime. Define the set A by
A = {b(F)u\u
> 0, Fe &}.
Property (a) of the Lemma implies that (b(Fx),b(F2),...
,b(Fh))>
1 for every Fx,
F2,...,Fh e &', and so (i) is satisfied. Property (b) of the Lemma implies that for
every prime p there are sets Fx,...,Fhe^r
such that Plf.ijF] = {p}, and so
(b(Fx), ...,b(Fh)) = p. It follows that the linear form b(Fx)ux + ■■■+b(Fh)uh
represents all sufficiently large multiples of p, and so np e hA for all n ^ N(p).
This proves (ii).
Theorem
3. There exists a set A of nonnegative integers such that (ax,...,
ah) > 1
for all ax,.. .,ah e A and hA has density 1.
Proof. Let A satisfy (i) and (ii) of Theorem 2. For any t e N, the sumset hA
contains all sufficiently large multiples of the primes /»,,..., p„ and so there exists a
constant c, such that
(M)(x)>4i-n(i-i))-c,
for all x. The divergence of the series £(1//j)
every e > 0 and x > x(e), and so d(hA) = 1.
implies that (hA)(x) > (1 — e)x for
Theorem 4. Let 2 < A < k. If A is a set of nonnegative integers such that
(ax,..., ak) > 1 for all ax,... ,ak e A, then the upper asymptotic density of the sumset
hA is strictly less than 1.
Proof.
density
Fix a* e A. Let B* = {n > 0\(n,a*)
> 1}. Then B* has asymptotic
d(B*)= \- n(i--M<i.
p\a*\
PI
Let n e hA. Then n = ax + ■■■ +ah for some ax,...,aheA.
Since k > h, it
follows that (ax,...,ah,
a*) = d > 1, and so d divides n for some divisor d > 1 of
a*. Therefore, hA ç B* and
du(hA) ^ d(B*) < 1.
Theorem 5. Let 2 < A < k. Let A be a set of nonnegative integers such that
(ax,..., ak) > 1 for all ax,...,ak
e A. Then A is not an asymptotic basis of order A.
In particular, if p is prime, then p £ hA.
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14
M. B. NATHANSON
Proof.
(ax,...,ah)
Let n e hA. Then n = ax + ••• +ah
= d^2. Then
n = ax+
■■■+ah = d¡[^+
for some ax,...,
■■■+^
ah e A, and
= dd',
where d' > A > 2. Thus, n e hA implies that « is composite. This concludes the
proof.
Note that Theorem 5 shows that Theroems 1 and 2 are best possible.
3. Open problems. 1. Let the set A of nonnegative integers satisfy GCD(A), and
let E = N \ hA. Theorem 5 shows that
E(x)
> tr(x) — x/\ogx.
There exist sets A such that E(x) < cx/\oglogx.
(For example, let h = 2 and
mXj = 2j — \ and m2j = 2j in the Lemma, and construct A as in the proof of
Theorem 2.) It is not known if there exists a set A satisfying GCD(A) such that
E(x) has order of magnitude jt/loglogx. Indeed, there is no example of a set A
satisfying GCD(A) such that limsup£(x)logjc/x
= oo, or even E(x) > 2x/logJc.
2. Let 2 < A < k and let e > 0. Does there exist a set A satisfying GCD(k) such
that d(hA) > 1 - e?
3. Let A be a set of nonnegative integers with (ax, a2) = 1 for all ax, a2 e A.
Clearly, A(x) < tr(x) and A contains at most one even integer, hence the odd
integers in 2A have density zero. If P is the set of primes, then {n e N \2n e 2P}
has density 1. Is it possible to construct a set A satisfying (ax, a2) = 1 for all ax,
a2e A such that 2A contains all sufficiently large even integers?
Department
of Mathematics, Rutgers University, Newark, New Jersey 07102
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