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

TL;DR

This paper extends the Erd ext{"o}s-Tetali theorem by constructing sequences with prescribed growth and representation properties, using regular variation theory, and explores applications to additive bases and Schnirelmann's method.

Contribution

It introduces new conditions under which sequences with specific additive representation functions can be constructed, expanding the understanding of additive bases.

Findings

Constructed sequences with growth rate (x) and specific representation functions.

Extended the Erd ext{"o}s-Tetali theorem to broader classes of functions.

Applied regular variation theory to additive number theory problems.

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.