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

Abstract

The set A = { a n } n = 1 ∞ 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 ∼ α n h for some real number α > 0 , then α 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 η h ⩽ 1 / h ! such that N ( h ) = ( 0 , η h ) or N ( h ) = ( 0 , η 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 } n = 1 ∞ such that b n ∼ 2 n and B contains a subsequence { b n k } k = 1 ∞ such that b n k ∼ 3 k .