An extension of the Erd\H{o}s-Tetali theorem

Given a sequence $\mathcal{A}=\{a_0<a_1<a_2\dots \}\subseteq \mathbb{N}$, let $r_{\mathcal{A},h}(n)$ denote the number of ways $n$ can be written as the sum of $h$ elements of $\mathcal{A}$. Fixing $h\ge 2$, we show that if $f$ is a suitable real function (namely: locally integrable, $O$-regularly varying and of positive increase) satisfying $$x^{1\mathrm{/}h}\log (x{)}^{1\mathrm{/}h}\ll f(x)\ll x^{1\mathrm{/}(h-1)-\epsilon }\text{ for some }\epsilon >0,$$ then there must exist $\mathcal{A}\subseteq \mathbb{N}$ with $\mathrm{\mid }\mathcal{A}\cap [0,x]\mathrm{\mid }=\mathrm{\Theta }(f(x))$ for which $r_{\mathcal{A},h+\mathrm{\ell }}(n)=\mathrm{\Theta }(f(n{)}^{h+\mathrm{\ell }}\mathrm{/}n)$ for all $\mathrm{\ell }\ge 0$. Furthermore, for $h=2$ the same conclusion holds under $x^{1\mathrm{/}2}\log (x{)}^{1\mathrm{/}2}\ll f(x)\ll x$. The proof is somewhat technical and the methods rely on ideas from regular variation theory, which are presented in an appendix with a view towards the general theory of additive bases. We also mention an application of these ideas to Schnirelmann’s method. Corrections to the published version are highlighted in red.