An extension of the Erdős-Tetali theorem

Abstract

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 \[ x^{1/h}\log(x)^{1/h} \ll f(x) \ll x^{1/(h-1) - \varepsilon} \text{ for some } \varepsilon > 0, \] then there must exist $\mathcal{A}\subseteq \mathbb{N}$ with $|\mathcal{A}\cap [0,x]|=\Theta (f(x))$ for which $r_{\mathcal{A},h+\ell }(n)=\Theta (f(n{)}^{h+\ell }/n)$ for all $\ell \ge 0$. Furthermore, for $h=2$ the same conclusion holds under $x^{1/2}\log (x{)}^{1/2}\ll f(x)\ll x$. 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. Corrections to the published version are highlighted in red.

38 pages. Removed Lemma 5.5 and the related argument due to a gap in Part 2 of its proof, resulting in a more restrictive hypothesis in the Main Theorem; several additional inaccuracies were corrected (changes in red). The full range stated in the published theorem follows by combining Theorem 1.4 of arXiv:2405.01530 with Theorem 5.6

References in corpus (1)

• Upper tails for arithmetic progressions in random subsets

Cited by in corpus (1)

• An elementary heuristic for Hardy-Littlewood extended Goldbach's conjecture