ON ASYMPTOTIC BASES WHICH HAVE DISTINCT SUBSET SUMS
Keyword(s):
Abstract 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.
1960 ◽
Vol 12
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pp. 374-389
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1991 ◽
Vol 14
(3)
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pp. 457-462
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1961 ◽
Vol 5
(1)
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pp. 35-40
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2018 ◽
Vol 11
(04)
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pp. 1850056
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2021 ◽
Vol 14
(2)
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pp. 380-395