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

TL;DR

This paper investigates the additive spectrum of order h in number theory, establishing bounds and properties of sequences that form asymptotic bases, using supersequence construction and sequence rearrangement techniques.

Contribution

It introduces bounds for the additive spectrum of order h and employs novel methods involving supersequences and sequence rearrangements in additive number theory.

Findings

The additive spectrum N(h) is either (0, eta_h) or (0, eta_h] with eta_h <= 1/h!

Constructs supersequences with prescribed asymptotic growth for additive bases.

Proves non-existence of certain sequence subsequences with specific asymptotic behaviors.

Abstract

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.