On minimal asymptotic basis of order g−1

Let ${\mathcal{A}}=\{a_{1}<a_{2}<\cdots \,\}$ be a set of nonnegative integers. Put $D({\mathcal{A}})=\gcd \{a_{k+1}-a_{k}:k=1,2,\ldots \}$. The set ${\mathcal{A}}$ is an asymptotic basis if there exists $h$ such that every sufficiently large integer is a sum of at most $h$ (not necessarily distinct) elements of ${\mathcal{A}}$. We prove that if the difference of consecutive integers of ${\mathcal{A}}$ is bounded, then ${\mathcal{A}}$ is an asymptotic basis if and only if there exists an integer $a\in {\mathcal{A}}$ such that $(a,D({\mathcal{A}}))=1$.

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DENSE SETS OF INTEGERS WITH A PRESCRIBED REPRESENTATION FUNCTION

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972711002358 ◽

2011 ◽

Vol 84 (1) ◽

pp. 40-43 ◽

Cited By ~ 3

Author(s):

MIN TANG

Keyword(s):

Real Number ◽

Representation Function ◽

Dense Sets ◽

Asymptotic Basis ◽

Positive Integers

AbstractA set A⊆ℤ is called an asymptotic basis of ℤ if all but finitely many integers can be represented as a sum of two elements of A. Let A be an asymptotic basis of integers with prescribed representation function, then how dense A can be? In this paper, we prove that there exist a real number c>0 and an asymptotic basis A with prescribed representation function such that $A(-x,x)\geq c\sqrt {x}$ for infinitely many positive integers x.

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Efficient asymptotic basis to reduce the forced dynamic problem of viscoelastic sandwich plates

MATEC Web of Conferences ◽

10.1051/matecconf/20168305003 ◽

2016 ◽

Vol 83 ◽

pp. 05003

Author(s):

F. Boumediene ◽

E.M. Daya ◽

J.M. Cadou ◽

L. Duigou

Keyword(s):

Dynamic Problem ◽

Sandwich Plates ◽

Asymptotic Basis

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Representation functions of additive bases for abelian semigroups

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171204306046 ◽

2004 ◽

Vol 2004 (30) ◽

pp. 1589-1597 ◽

Cited By ~ 5

Author(s):

Melvyn B. Nathanson

Keyword(s):

Large Class ◽

Abelian Semigroup ◽

Representation Function ◽

Asymptotic Basis ◽

Countably Infinite ◽

The Given

A subset of an abelian semigroup is called an asymptotic basis for the semigroup if every element of the semigroup with at most finitely many exceptions can be represented as the sum of two distinct elements of the basis. The representation function of the basis counts the number of representations of an element of the semigroup as the sum of two distinct elements of the basis. Suppose there is given function from the semigroup into the set of nonnegative integers together with infinity such that this function has only finitely many zeros. It is proved that for a large class of countably infinite abelian semigroups, there exists a basis whose representation function is exactly equal to the given function for every element in the semigroup.

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On g-adic minimal asymptotic bases of order h

International Journal of Number Theory ◽

10.1142/s1793042119500209 ◽

2019 ◽

Vol 15 (02) ◽

pp. 389-406 ◽

Cited By ~ 1

Author(s):

Cui-Fang Sun ◽

Tian-Tian Tao

Keyword(s):

Nonempty Subset ◽

Asymptotic Basis

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].

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A NOTE ON ASYMPTOTIC NONBASES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972716000678 ◽

2016 ◽

Vol 95 (1) ◽

pp. 1-4

Author(s):

DENG-RONG LING

Keyword(s):

Asymptotic Basis

Let $A$ be a subset of $\mathbb{N}$, the set of all nonnegative integers. For an integer $h\geq 2$, let $hA$ be the set of all sums of $h$ elements of $A$. The set $A$ is called an asymptotic basis of order $h$ if $hA$ contains all sufficiently large integers. Otherwise, $A$ is called an asymptotic nonbasis of order $h$. An asymptotic nonbasis $A$ of order $h$ is called a maximal asymptotic nonbasis of order $h$ if $A\cup \{a\}$ is an asymptotic basis of order $h$ for every $a\notin A$. In this paper, we construct a sequence of asymptotic nonbases of order $h$ for each $h\geq 2$, each of which is not a subset of a maximal asymptotic nonbasis of order $h$.

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