Let $h,k \ge 2$ be integers. We say a set $A$ 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 $A$. A set of positive integers $A$ is called $B_{h}[g]$ set if all positive integers can be represented as the sum of $h$ terms from $A$ at most $g$ times. In this paper we prove the existence of $B_{h}[1]$ sets which are asymptotic bases of order $2h+1$ by using probabilistic methods.

中文翻译：

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广义渐近 Sidon 基

令 $h,k \ge 2$ 为整数。如果每个足够大的正整数都可以表示为来自 $A$ 的 $k$ 项的总和，我们说正整数的集合 $A$ 是 $k$ 阶的渐近基。一组正整数 $A$ 称为 $B_{h}[g]$ 集合，如果所有正整数都可以表示为 $A$ 至多 $g$ 次的 $h$ 项的总和。在本文中，我们使用概率方法证明了$B_{h}[1]$ 集的存在，它是$2h+1$ 阶的渐近基。