ON ASYMPTOTIC BASES WHICH HAVE DISTINCT SUBSET SUMS

Let k and l be positive integers satisfying $k \ge 2, l \ge 1$ . A set $\mathcal {A}$ of positive integers is an asymptotic basis of order k if every large enough positive integer can be represented as the sum of k terms from $\mathcal {A}$ . About 35 years ago, P. Erdős asked: does there exist an asymptotic basis of order k where all the subset sums with at most l terms are pairwise distinct with the exception of a finite number of cases as long as $l \le k - 1$ ? We use probabilistic tools to prove the existence of an asymptotic basis of order $2k+1$ for which all the sums of at most k elements are pairwise distinct except for ‘small’ numbers.

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].

A construction of maximal asymptotic nonbases

Let [Formula: see text] denote the set of all nonnegative integers and [Formula: see text] be a subset of [Formula: see text]. The set [Formula: see text] is called an asymptotic basis of order [Formula: see text] if every sufficiently large integer can be written as the sum of two elements of [Formula: see text]. Otherwise, [Formula: see text] is called an asymptotic nonbasis of order [Formula: see text]. Let [Formula: see text] denote the number of representations of [Formula: see text] in the form [Formula: see text], where [Formula: see text] and [Formula: see text]. An asymptotic nonbasis [Formula: see text] of order [Formula: see text] is called a maximal asymptotic nonbasis of order [Formula: see text] if [Formula: see text] is an asymptotic basis of order [Formula: see text] for every [Formula: see text]. In this paper, a maximal asymptotic nonbasis [Formula: see text] is constructed satisfying [Formula: see text] for all [Formula: see text] and [Formula: see text] as [Formula: see text], where [Formula: see text] is an increasing sequence of [Formula: see text].

A NOTE ON THE ERDŐS–GRAHAM THEOREM

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 $(a,D({\mathcal{A}}))=1$.

A NOTE ON ASYMPTOTIC NONBASES

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 $h$.

DENSE SETS OF INTEGERS WITH A PRESCRIBED REPRESENTATION FUNCTION

A 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.