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

Melvyn B. Nathanson

Arxiv ID: 0806.0984•Last updated: 1/6/2021

The set A = {a_n} 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 a_n ~ alpha n^h 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 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 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 = {b_n} such that b_n ~
2^n and B contains a subsequence {b_{n_k}} such that b_{n_k} ~ 3^k.

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