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The set A = {a_n} of nonnegative integers is an asymptotic basis of order hif every sufficiently large integer can be represented as the sum of h elementsof A. If a_n ~ alpha n^h for some real number alpha > 0, then alpha is calledan additive eigenvalue of order h. The additive spectrum of order h is the setN(h) consisting of all additive eigenvalues of order h. It is proved that thereis a positive number eta_h <= 1/h! such that N(h) = (0, eta_h) or N(h) = (0,eta_h]. The proof uses results about the construction of supersequences ofsequences with prescribed asymptotic growth, and also about the asymptotics ofrearrangements of infinite sequences. For example, it is proved that there doesnot exist a strictly increasing sequence of integers B = {b_n} such that b_n ~2^n and B contains a subsequence {b_{n_k}} such that b_{n_k} ~ 3^k.

Supersequences, rearrangements of sequences, and the spectrum of bases in additive number theory