A set A = Ak,n ? [n] [ {0} is said to be an additive k-basis if each element in {0, 1, . . . , kn} can be written as a k-sum of elements of A in at least one way. Seeking multiple representations as k-sums, and given any function (n) ! 1, we say that A is said to be a truncated (n)-representative k-basis for [n] if for each j 2 [?n,(k ?)n] the number of ways that j can be represented as a k-sum of elements of Ak,n is ?((n)). In this paper, we follow tradition and focus on the case (n) = log n, and show that a randomly selected set in an appropriate probability space is a truncated log-representative basis with probability that tends to one as n ! 1. This result is a finite version of a result proved by Erd?os and extended by Erd?os and Tetali.
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