Some remarks on minimal bases and maximal nonbases of integers

Discrete Mathematics North-Holland

122 (1993) 357-362

357

Note

Some remarks on minimal bases and maximal nonbases of integers Xing-De Department

Jia of’ Mathematics,

Southwest

Te.uus State

UKuersity.

San Murcos,

TX 78666.

USA

Received 17 September 1990 Revised 8 October 1991

Abstract In this note, we generalize the concepts of minimal bases and maximal nonbases prove some existence theorems for the generalized minimal bases and maximal generalize some results of Stiihr, Deza and Erdds, and Nathanson.

for integers, and nonbases, which

1. Introduction Let N denote

the set of nonnegative integers. h for R4 if every large integer is elements of A. An asymptotic basis A of order h an asymptotic basis of order h. It is important

basis of order

A set A of N is called an nsymptotic a sum of h not necessarily distinct is minimal if no proper subset of A is to notice that not every asymptotic

basis of order h contains a minimal asymptotic basis of order h. A trivial example is A={l,h,2h,3h ,... }, which is an asymptotic basis of order h containing no minimal asymptotic basis of order h. The set of all squares is a basis of order four. It is not known if there exists a minimal basis of order four containing only squares. This concept of minimality of bases was first introduced by Stiihr [l 11. Hsrtter [6] showed the existence of minimal asymptotic bases by using a nonconstructive argument. Nathanson [7] constructed the first nontrivial example of an asymptotic basis of order two, no subset of which is a minimal asymptotic basis of order two. Furthermore, ErdGs and Nathanson [2] constructed a family of asymptotic bases A of order two such that A\S is an asymptotic basis of to: Xing-De TX 18666, USA.

Correspondence

Marcos,

0012-365X/93/$06.00

Jia, Department

(0 1993-Elsevier

of Mathematics,

Science Publishers

Southwest

Texas State University,

B.V. All rights reserved

San

35x

X.-D.

Jicr

order two if and only if S is a finite subset of A. Therefore, A does not contain any minimal asymptotic basis of order two. Erdiis and Nathanson [3] found a simple condition that an asymptotic basis of order two contains a minimal asymptotic basis of order two. When h 3 3, it is an unsolved problem to determine basis of order h contains a minimal asymptotic basis of order h.

if an asymptotic

There are some generalizations of bases and asymptotic bases for integers. One interesting case is to replace the set of integers by the collection of all finite subsets of integers, and the operation of addition by the set-theoretic union of sets. Deza and ErdBs [I] proved analogues of the Erdbs-Landau Theorem and Schnirelmann’s Theorem. Nathanson [S] and Grekos [4,5] studied minimal union bases and maximal union nonbases. Nathanson [lo] also studied multiplicative bases for integers. In this note, we shall generalize the concept of minimal bases and maximal nonbasis for integers, and prove some existence theorems for the generalized minimal bases and maximal nonbases. Let h be a positive integer. Let M and N be two nonempty sets, Mh= M x M x ... x M (h times) the Cartesian product. Let 6: M” -+N be an onto mapping. A subset A EM is called a hasis of order h for N (under cr) if a( A*) = N. A basis A of order h is minimal if no proper subset of A is a basis of order h. A subset AE M is called a nonhasis of order h for N (under a) if A is not a basis of order h. Nathanson [7] introduced maximal nonbases, the dual concept of minimal bases. A nonbasis A of order h is called a maxima/ nonhasis c$“order h if Au i a) is a basis of order h for every u#A. Similarly, we may define the asymptotic version of all these concepts. For instance, a subset A c M is called an asymptotic basis of order h if N\o( A”) is finite. These concepts are generalizations of that for integers. For instance, if we take M = N to be the set of all nonnegative integers, and o(ar , . . . , ah) = aI + ... + a,,, then the bases and minimal bases are the original bases and minimal bases for integers. If M = N is the set of positive integers, and o(ar , . . , &)=l.c.m.(ur , . . . ,u,,), the least common multiple of a,, . . , ah, then the bases and minimal bases are the LCM bases and LCM minimal bases defined by Nathanson [lo]. Let M = N be the set of all finite the set theoretical subsets of the set of all integers, and o( A,, . . . , A,,)= Alu...uAh, union of sets, then the bases for this N are the bases of Deza and Erdiis [l] and studied by others. Let M and N denote two countably infinite sets. Let A be a subset of M. For any element UEN, S( A, h, u) denotes the collection of the elements ( ul, . . . , u&Ah such that u=o(~~,...,a,).Denoter(A,h,u)=IS(A,h,u)(.Foranya~S(A,h,u),S(a)denotesthe set consisting of all the h coordinates of a. We use hA to denote a(Ah).

2. Results Theorem 1. Lrt A be a basis of order A contains

a minimal

basis

of order

h.

h for

N. If r( A, h, u) < CC jbr

every

UE M, then

Some remarks on minimal bases and ma.wimalnonbases of integers

359

Proof. Suppose that A = {al, u2, . . . } is not a minimal basis of order h. Then A\{ ai} is also a basis of order h for some Ui~A. Let ii be the least such integer. If A, is minimal then we are done. Otherwise, let iz be the least integer such that A, = A,\{ ui2} is also a basis of order h. Continue this procedure inductively. If it stops in finitely many steps, say, at Ako. Then Ako is a minimal basis of order h which is contained in A. If it does not stop in finitely many steps, we have the following infinite decreasing sequence of bases of order h:

where A,=A,_,\{ai*) A=

{

for k=l,2,...,

and i,
Let

A,=A\{a,Ik=1,2,...}.

k=l

We shall prove that A is a minimal basis of order h. Let UE N. Since Ak is a basis of order h, S(A,, h, u) # 8. Hence we have the following decreasing sequence:

Noticing that

that 1S( A, h, u) 1=r( A, h, u) < CD, we see that there exists an integer

k,, such

Therefore, S( A, h, u) = S( Ako, h, u) # 0, which means that u~hA. Hence A is a basis of order h. If A \ { ai} is also a basis of order h for some Ui~A, then there exists an integer k such thati,_,
u (S(a)\A,) oeS(M,h. UO)

contains at least one coordinate of each element a of S( M, h, u,), where S(a) denotes the set consisting of all the h coordinates of a. Noticing that u$A, we see that X1 is

360

x.-n. .licr

a proper subset of X= M\A, which contradicts the minimality of X. Therefore, A is a maximal nonbasis of order h. Conversely, let A be a maximal nonbasis of order h. It is clear that X= M\A contains at least one coordinate of each element in S( M, h, u) for every usN\hA. If, for some u,EN\~A and some UEX, i.e., a$A, X’=X\(a} also contains at least one coordinate of each element ~ES( M. h, uo), then (M\( Au{ a}))nS(a)#@ ~ES(M, 11,uO). This implies that u,$h( Au (u] ). Hence, A is not a maximal order h. The proof is complete. 0 Theorem 3. !fr( M, h, u)< z,ftir a maximal nonhasis of order h.

all UFN, then every nonhusis

for every nonbasis of

qf’order h is contained

in

Proof. Let A be a nonbasis of order h. Then there exists ue~N such that u,#hA. Hence S(a)$A for any UES( M, h,uo). Let X be a minimal set for which 0 # XnS(a)cS(a)\A for all UES( M, h, q,). The set X exists because S( M, h, uo) is finite. It is clear that AI = M \,X is a nonbasis of order h. It follows from the finiteness of S( M, h, u,,) that X is a finite set with at most r( M, h, u,,) elements. Therefore, there exists a maximal nonbasis A of order h containing A,, hence, 22 A. The proof is complete. U The following results Theorems 1 and 3.

of Stohr

[l l] and

Nathanson

Corollary 2. Every basis of order h,for nonnegative

integers

order

integers

h. Every

nonhasis

nonhasis

(If‘ order

qf order

hfor

nonnegative

[7]

are immediate

contains

u minimal

is contained

from

basis of

in a muximul

h.

A set A of positive integers is an LCM hasis of order h if every positive integer is the least common multiple of h not necessarily distinct elements in A. This concept was introduced by Nathanson [IS]. Corollary 3. Every LCM

LCM

husis of‘ order

husis of order h. Every

nonhasis

@order

h,for

nonhusis

the positive

integers

of order h is contained

contains in a LCM

a minimal maximal

h.

In what follows, we o(ar, . . ..q$)=ar “‘Uh. Theorem 4. Suppose

assume

S is (I subset

representation

as a product

qfS.

denote

Let (Si)

LCM

of.finitely

the suhmonoid

that

M=N

is a commutative

of M such that every many elements generated

element

in S. Let { Sr , .

monoid,

and

qf M has only one

, S,} he a partition

by Si. !f there ure ut least two (Si)

are

361

injinite, then

is a minimal asymptotic

basis of order h for M.

Proof. Let UEM. Then u=al...a, a partition of S, we may assume

for some a+S (i=l,...,n). that

Since {S,,...,S,}

is

This implies that uehA. Hence, A is an asymptotic basis of order h. Let UEA, say, u=ur~(Sr). Since every element in A4 has a unique representation a sum of elements in S, it follows that

as

a,ESr ...

for ,~=l,...,i,

a,sSh

for p=iih_l+l,...,n.

. ...

Hence, ii

a,E(S,),...,

p=l

fi

a,C(Sh).

p=ih-,+l

A,=

fi UiIUi~(Si)

for i=2,...,h

i=l

is infinite because at least one of (Si) is infinite. Clearly A,nh(A\ { u})=@, hence A\{ u} is not an asymptotic basis of order h. Therefore, A is a minimal asymptotic basis of order h. The proof is complete. 0

Acknowledgement The author

[l] [2] [3]

[4] [5] [6]

likes to thank

the referees for their helpful suggestions

and comments.

M. Deza and P. Erdiis, Extension de quelques theoremes sur les densities de series d’elements de N a des series de sousensembles finis de N, Discrete Math. I2 (1975) 2955308. P. Erdiis and M.B. Nathanson, Oscillations of bases for the natural numbers, Proc. Amer. Math. Sot. 35 (1975) 253-258. P. Erdds and M.B. Nathanson, Systems of distinct representatives and minimal bases in additive number theory, in: M.B. Nathanson ed.. Proceedings, Number Theory, Carbondale 1979, Lecture Notes in Mathematics, Vol. 751 (Springer, Berlin, 1979) 89-107. G. Grekos, Quelques Aspects de la Theorie Additive des Nombres, Thesis, Universite de Bordeaux I, 1982. G. Grekos, Nonexistence of maximal asymptotic union nonbases, Discrete Math. 33 (1981) 2677270. E. Hartter, Ein Beitrag zur Theorie der Minimalbasen, J. Reine Angew. Math. 196 (1956) 170-204.

362

[7]

X.-D. Jiu

M.B. Nathanson, Minimal bases and maximal nonbases in additive number theory, J. Number Theory 6 (1974) 3244333. [S] M.B. Nathanson, Oscillations of bases in number theory and combinatorics, Number Theory Day, Lecture Notes in Mathematics, Vol. 626 (Springer, Berlin, 1977) 217-231. [9] M.B. Nathanson, Multiplicative representation of integers, Israel J. Math. 57 (1987) 1299136. [lo] M.B. Nathanson, An extremal problem for least common multiples, Discrete Math. 64 (1987) 221-228. [1 1] A. Stohr, Geliiste und ungeliiste Fragen iiber Basen der natiirlichen Zahlenreiht, 11, J. Reine Angew. Math. 194 (1955) 11 l-140.