For a cyclic group G and a ∈ G, we define a as the subgroup generated by a and the atom of a as the set of all elements generating a . In this paper, given t (t ≥ 2) atoms, we obtain an exact formula for the number of representations of each element in the sumset of these t atoms. We also show explicitly which atoms are part of the union which constitutes the sumset of t given atoms. The case t = 2 was recently obtained by Sander and Sander.

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Let [Formula: see text] be the set of all nonnegative integers. Let [Formula: see text] be a subset of [Formula: see text] and [Formula: see text] be a nonempty subset of [Formula: see text]. Denote by [Formula: see text] the set of all finite, nonempty subsets of [Formula: see text]. For integer [Formula: see text], let [Formula: see text] be the set of all numbers of the form [Formula: see text], where [Formula: see text] and [Formula: see text]. Let [Formula: see text] be any integer. For [Formula: see text], let [Formula: see text]. In this paper, we show that for any [Formula: see text], the set [Formula: see text] is a minimal asymptotic basis of order [Formula: see text]. We also prove that for any [Formula: see text] and [Formula: see text], the set [Formula: see text] is a minimal asymptotic basis of order [Formula: see text].

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