Minimal Asymptotic Bases Research Articles

On a problem of Nathanson on minimal asymptotic bases

Let N denote the set of all nonnegative integers and A be a subset of N. Let h be an integer with h≥2. Let n∈N and rh(A,n)=♯{(a1,…,ah)∈Ah:a1+⋯+ah=n}. The set A is called an asymptotic basis of order h if rh(A,n)≥1 for all sufficiently large integer n. An asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order h. In 1988, Nathanson posed a problem on minimal asymptotic bases of order h. Recently, Chen and Tang showed that the answer to the problem is negative for h≥4 by constructing a special partition of N. In this paper, we give a new construction of minimal asymptotic bases. This construction expands our understanding on the problem of Nathanson.

SOME REMARKS ON MINIMAL ASYMPTOTIC BASES OF ORDER THREE

We study a question on minimal asymptotic bases asked by Nathanson [‘Minimal bases and powers of 2’, Acta Arith. 49 (1988), 525–532].

On g-adic minimal asymptotic bases of order h

Let [Formula: see text] be the set of all nonnegative integers. Let [Formula: see text] be a subset of [Formula: see text] and [Formula: see text] be a nonempty subset of [Formula: see text]. Denote by [Formula: see text] the set of all finite, nonempty subsets of [Formula: see text]. For integer [Formula: see text], let [Formula: see text] be the set of all numbers of the form [Formula: see text], where [Formula: see text] and [Formula: see text]. Let [Formula: see text] be any integer. For [Formula: see text], let [Formula: see text]. In this paper, we show that for any [Formula: see text], the set [Formula: see text] is a minimal asymptotic basis of order [Formula: see text]. We also prove that for any [Formula: see text] and [Formula: see text], the set [Formula: see text] is a minimal asymptotic basis of order [Formula: see text].

On minimal asymptotic bases of order three

Let $A$ be a subset of $\mathbb {N}$, and $W$ be a nonempty subset of $\mathbb {N}$. Denote by $\mathcal {F}^{\ast }(W)$ the set of all finite, nonempty subsets of $W$. For integer $g\geq 2$, let $A_{g}(W)$ be the set of all numbers of the form $ \sum _{f

On a problem of Nathanson

A set $A$ of nonnegative integers is an asymptotic basis of order $h$ if every sufficiently large integer can be represented as the sum of $h$ integers (not necessarily distinct) of $A$. An asymptotic basis $A$ of order $h$ is minimal if no proper subset of $A$ is an asymptotic basis of order $h$. In this paper, we resolve a problem of Nathanson on minimal asymptotic bases of order $h$.

On minimal asymptotic bases

Let N denote the set of all nonnegative integers. Let W be a nonempty subset of N. Denote by F∗(W) the set of all finite, nonempty subsets of W. Let A(W) be the set of all numbers of the form ∑f∈F2f, where F∈F∗(W). Let N=W1∪W2 be a partition with 0∈W1 such that W1 and W2 are infinite. In this paper, we prove that A=A(W1)∪A(W2) is a minimal asymptotic basis of order 2 if and only if either W1 contains no consecutive integers or W2 contains consecutive integers or both. We resolve three problems on asymptotic bases of order 2 which had been posed by Nathanson.

Dense minimal asymptotic bases of order two

We call a set A of positive integers an asymptotic basis of order h if every sufficiently large integer n can be written as a sum of h elements of A. If no proper subset of A is an asymptotic basis of order h, then A is a minimal asymptotic basis of that order. Erdős and Nathanson showed that for every h ⩾ 2 there exists a minimal asymptotic basis A of order h with d ( A ) = 1 / h , where d ( A ) denotes the density of A. Erdős and Nathanson asked whether it is possible to strengthen their result by deciding on the existence of a minimal asymptotic bases of order h ⩾ 2 such that A ( k ) = k / h + O ( 1 ) . Moreover, they asked if there exists a minimal asymptotic basis with lim sup ( a i + 1 − a i ) = 3 . In this paper we answer these questions in the affirmative by constructing a minimal asymptotic basis A of order 2 fulfilling a very restrictive condition 1 2 k ⩽ A ( k ) ⩽ 1 2 k + 1 .

On the Order of a Minimal Additive Basis

LetAbe a set of non-negative integers. If every sufficiently large integer is the sum ofhnot necessarily distinct elements ofA, thenAis called anasymptotic basis of order h. An asymptotic basisAof orderhis called minimal if no proper subset ofAis an asymptotic basis of orderh. It is proved that for every integerh⩾3, no setAis simultaneously a minimal asymptotic basis of ordershand 2h. Forh=2, the problem had already been solved.

Independence of solution sets and minimal asymptotic bases

Independence of solution sets and minimal asymptotic bases

A construction of minimal asymptotic bases

Jia and Nathanson [2] give a simple and explicit construction of minimal asymptotic bases of orderh for everyh≥2. They constructed minimal asymptotic bases from partition of N by means of powers of 2. In this paper, we extend the results of that paper to asymptotic bases constructed from partitions of N by means ofg-adic representations forg≥2. Corollary 3 shows that given partition N=W0⌣W1⌣...⌣Wh−1 such that eachWi contains infinitely many pairs of consecutive integers we can construct a minimal asymptotic bases of orderh in infinitely many way.

Simultaneous systems of representatives and combinatorial number theory

This paper applies a recent theorem about simultaneous systems of representatives for families of finite sets to obtain combinatorial results that are closely related to problems concerning minimal asymptotic bases in additive number theory.

On the maximum density of minimal asymptotic bases

A set A of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer is the sum of h elements of A . It is proved that if A is an asymptotic basis of order h with lower asymptotic density dL (A) > /h , then there is a set W contained in A such that W has positive asymptotic density and A\W is an asymptotic basis of order h . This implies that if A is a minimal asymptotic basis of order h , then dL (A) 2. Minimal asymptotic bases form an important extremal class in additive number theory. The asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order h. It follows that if A is minimal, then for every element a E A there must be infinitely many positive integers n, each of whose representations as a sum of h elements of A includes the number a as a summand. St6hr [7] introduced the concept of minimal asymptotic basis, and Hartter [3] proved that minimal asymptotic bases of order h exist for all h > 2. Erdos and Nathanson [1] survey recent results on minimal asymptotic bases. For any set A of integers, the counting function of A, denoted A(x), is defined by A(x) = card({a E Al 2, Nathanson [5, 6] has constructed minimal asymptotic bases A of order h that satisfy A(x) x for every Received by the editors December 12, 1986 and, in revised form, January 11, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 11 B 1 3, 1 1 B05, 11 P99.

On the Maximum Density of Minimal Asymptotic Bases

On the Maximum Density of Minimal Asymptotic Bases

Minimal asymptotic bases with prescribed densities

Minimal asymptotic bases with prescribed densities

Sets of Natural Numbers with no Minimal Asymptotic Bases

Sets of Natural Numbers with no Minimal Asymptotic Bases

Sets of natural numbers with no minimal asymptotic bases

The set A of natural numbers is an asymptotic basis for S if the sets S and 2A eventually coincide. An asymptotic basis A for S is minimal if no proper subset of A is a basis for S. Sets S are constructed which possess infinitely many asymptotic bases, but no minimal asymptotic basis.

Minimal bases and maximal nonbases in additive number theory

An asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order h. Examples are constructed of minimal asymptotic bases, and also of an asymptotic basis of order two no subset of which is minimal. If B is a set of nonnegative integers which is not a basis (resp. asymptotic basis) of order h, but such that every proper superset of B is a basis (resp. asymptotic basis) of order h, then B is a maximal nonbasis (resp. maximal asymptotic nonbasis) of order h. Examples of such sets are constructed, and it is proved that every set not a basis of order h is a subset of a maximal nonbasis of order h.