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.

中文翻译：

---

基于具有不同子集和的渐近基

让ķ和l是满足的正整数$k \ge 2, l \ge 1$. 一套$\数学{A}$正整数的一个渐近基序ķ如果每个足够大的正整数都可以表示为ķ条款来自$\数学{A}$. 大约 35 年前，P. Erdős 问道：是否存在有序的渐近基础ķ其中所有子集最多与l除了有限数量的情况外，术语是成对不同的，只要$l \le k - 1$? 我们使用概率工具来证明存在一个渐近序基$2k+1$其中所有的总和最多ķ除了“小”数字外，元素是成对不同的。