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ON ASYMPTOTIC BASES WHICH HAVE DISTINCT SUBSET SUMS

Published online by Cambridge University Press:  12 April 2021

SÁNDOR Z. KISS
Affiliation:
Institute of Mathematics, Budapest University of Technology and Economics, H-1529 B.O. Box, Budapest, Hungary
VINH HUNG NGUYEN
Affiliation:
Institute of Mathematics, Budapest University of Technology and Economics, H-1529 B.O. Box, Budapest, Hungary e-mail: nguyenvinhhung108@gmail.com
Corresponding

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.

Type
Research Article
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

The first author was supported by the National Research, Development and Innovation Office NKFIH Grant Nos. K115288 and K129335. This paper was supported by a János Bolyai Research Scholarship of the Hungarian Academy of Sciences, by the ÚNKP-19-4 New National Excellence Program of the Ministry for Innovation and Technology and by the ÚNKP-20-5 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund.

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ON ASYMPTOTIC BASES WHICH HAVE DISTINCT SUBSET SUMS
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