Abstract
AbstractLet 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.
Publisher
Cambridge University Press (CUP)