On Waring's problem: Beyond Freĭman's theorem

Abstract

AbstractLet satisfy . Freĭman's theorem shows that when , there exists such that all large integers are represented in the form , with , if and only if diverges. We make this theorem effective by showing that, for each fixed , it suffices to impose the condition More is established when the sequence of exponents forms an arithmetic progression. Thus, for example, when and , all large integers are represented in the form , with .