An extension of the Erdős-Tetali theorem
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866917870255996928 |
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| author | Táfula, Christian |
| author_facet | Táfula, Christian |
| contents | 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 $|\mathscr{A}\cap [0,x]|=Θ(f(x))$ for which $r_{\mathscr{A},h+\ell}(n) = Θ(f(n)^{h+\ell}/n)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1807_10200 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | An extension of the Erdős-Tetali theorem Táfula, Christian Combinatorics Number Theory 11B13, 11B34 (Primary), 05D40 (Secondary) 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 $|\mathscr{A}\cap [0,x]|=Θ(f(x))$ for which $r_{\mathscr{A},h+\ell}(n) = Θ(f(n)^{h+\ell}/n)$ 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. |
| title | An extension of the Erdős-Tetali theorem |
| topic | Combinatorics Number Theory 11B13, 11B34 (Primary), 05D40 (Secondary) |
| url | https://arxiv.org/abs/1807.10200 |