On asymptotic bases which have distinct subset sums

Kiss, Sándor and Nguyen, Vinh Hung (2021) On asymptotic bases which have distinct subset sums. BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY, 104 (2). pp. 211-217. ISSN 0004-9727

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Abstract

Let k and l be positive integers satisfying k≥2,l≥1 . 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 . 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≤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.

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