Supersequences, rearrangements of sequences, and the spectrum of bases in additive number theory
Abstract
The set A = an of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer can be represented as the sum of h elements of A. If an ~ alpha nh for some real number alpha > 0, then alpha is called an additive eigenvalue of order h. The additive spectrum of order h is the set N(h) consisting of all additive eigenvalues of order h. It is proved that there is a positive number etah <= 1/h! such that N(h) = (0, etah) or N(h) = (0, etah]. The proof uses results about the construction of supersequences of sequences with prescribed asymptotic growth, and also about the asymptotics of rearrangements of infinite sequences. For example, it is proved that there does not exist a strictly increasing sequence of integers B = bn such that bn ~ 2n and B contains a subsequence bnk such that bnk ~ 3k.