Cross representations of additive complements of $r$-th powers
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| Format: | Preprint |
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2025
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| _version_ | 1866917508273930240 |
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| author | Ding, Yuchen Sándor, Csaba Zhang, Zihan |
| author_facet | Ding, Yuchen Sándor, Csaba Zhang, Zihan |
| contents | Let $\mathbb{N}$ be the set of natural numbers and $\mathcal{S}_r=\big\{1^r, 2^r, 3^r,\cdots\big\}$ the set of $r$-th powers, where $r\ge 2$ is a natural number. Let $\mathcal{W}_r$ be an additive complement of $\mathcal{S}_r$ and $$ f_r(n)=\#\big\{(w,m^r)\in \mathcal{W}\times \mathcal{S}_r: n=w+m^r\big\}. $$ Motivated by a 1993 conjecture of Cilleruelo, we show that $$ \sum_{n\le N}f_r(n)-N\gg_r N^{1-\frac{1}{r}}. $$ Previously, the bound was only proved for $r=2$. In the case $r=2$, the lower bound above can be made more explicit as $$ \sum_{n\le N}f_2(n)-N\gg N^{1/2}(\log N)^δ $$ for some absolute constant $δ>0$, which improves a $\log$ factor upon a recent result of Ding, Sun, Wang and Xia. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_15407 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cross representations of additive complements of $r$-th powers Ding, Yuchen Sándor, Csaba Zhang, Zihan Number Theory Let $\mathbb{N}$ be the set of natural numbers and $\mathcal{S}_r=\big\{1^r, 2^r, 3^r,\cdots\big\}$ the set of $r$-th powers, where $r\ge 2$ is a natural number. Let $\mathcal{W}_r$ be an additive complement of $\mathcal{S}_r$ and $$ f_r(n)=\#\big\{(w,m^r)\in \mathcal{W}\times \mathcal{S}_r: n=w+m^r\big\}. $$ Motivated by a 1993 conjecture of Cilleruelo, we show that $$ \sum_{n\le N}f_r(n)-N\gg_r N^{1-\frac{1}{r}}. $$ Previously, the bound was only proved for $r=2$. In the case $r=2$, the lower bound above can be made more explicit as $$ \sum_{n\le N}f_2(n)-N\gg N^{1/2}(\log N)^δ $$ for some absolute constant $δ>0$, which improves a $\log$ factor upon a recent result of Ding, Sun, Wang and Xia. |
| title | Cross representations of additive complements of $r$-th powers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.15407 |