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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2304.08694 |
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| _version_ | 1866915066350141440 |
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| author | Táfula, Christian |
| author_facet | Táfula, Christian |
| contents | Let $A\subseteq \mathbb{Z}_{\geq 0}$ be a finite set with minimum element $0$, maximum element $m$, and $\ell$ elements strictly in between. Write $(hA)^{(t)}$ for the set of integers that can be written in at least $t$ ways as a sum of $h$ elements of $A$. We prove that $(hA)^{(t)}$ is "structured" for
\[ h \geq (1+o(1)) \frac{1}{e} m\ell t^{1/\ell} \]
(as $\ell \to \infty$, $t^{1/\ell} \to \infty$), and prove a similar theorem on the size and structure of $A\subseteq \mathbb{Z}^d$ for $h$ sufficiently large. Moreover, we construct a family of sets $A = A(m,\ell,t)\subseteq \mathbb{Z}_{\geq 0}$ for which $(hA)^{(t)}$ is not structured for $h\ll m\ell t^{1/\ell}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_08694 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the size and structure of $t$-representable sumsets Táfula, Christian Combinatorics Number Theory 11D07, 05A17 Let $A\subseteq \mathbb{Z}_{\geq 0}$ be a finite set with minimum element $0$, maximum element $m$, and $\ell$ elements strictly in between. Write $(hA)^{(t)}$ for the set of integers that can be written in at least $t$ ways as a sum of $h$ elements of $A$. We prove that $(hA)^{(t)}$ is "structured" for \[ h \geq (1+o(1)) \frac{1}{e} m\ell t^{1/\ell} \] (as $\ell \to \infty$, $t^{1/\ell} \to \infty$), and prove a similar theorem on the size and structure of $A\subseteq \mathbb{Z}^d$ for $h$ sufficiently large. Moreover, we construct a family of sets $A = A(m,\ell,t)\subseteq \mathbb{Z}_{\geq 0}$ for which $(hA)^{(t)}$ is not structured for $h\ll m\ell t^{1/\ell}$. |
| title | On the size and structure of $t$-representable sumsets |
| topic | Combinatorics Number Theory 11D07, 05A17 |
| url | https://arxiv.org/abs/2304.08694 |