The spectrum of bases of finite order in additive number theory

The set A of nonnegative integers is called a basis of order h if every nonnegative integer can be represented as the sum of h not necessarily distinct elements of A. The positive real number c is called an additive eigenvalue of order h if there is a basis A = \{a_i\} of order h such that a_i \sim c i^h. The set of all additive eigenvalues of order h is called the spectrum of order h, and denoted N(h). It is proved that \sup N(h) = \eta(h) \leq h!/ \Gamma^h(1+1/h), and that the spectrum N(h) is an interval of the form (\,0,\eta(h)\,) or (\,0,\eta(h)\,].

Mel Nathanson