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

Abstract

Given a sequence $\mathscr{A}=\{a_0<a_1<a_2\ldots\}\subseteq \mathbb{N}$, let $r_{\mathscr{A},h}(n)$ denote the number of ways $n$ can be written as the sum of $h$ elements of $\mathscr{A}$. Fixing $h\geq 2$, we show that if $f$ is a suitable real function (namely: locally integrable, $O$-regularly varying and of positive increase) satisfying \[ x^{1/h}\log(x)^{1/h} \ll f(x) \ll \frac{x^{1/(h-1)}}{\log(x)^{\varepsilon}} \text{ for some } \varepsilon > 0, \] then there must exist $\mathscr{A}\subseteq\mathbb{N}$ with $|\mathscr{A}\cap [0,x]|=\Theta(f(x))$ for which $r_{\mathscr{A},h+\ell}(n) = \Theta(f(n)^{h+\ell}/n)$ for all $\ell \geq 0$. Furthermore, for $h=2$ this condition can be weakened to $x^{1/2}\log(x)^{1/2} \ll f(x) \ll x$. 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.