Arithmetical structure of sumset intersections
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914395497431040 |
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| author | Marques, Diego Nathanson, Melvyn B. |
| author_facet | Marques, Diego Nathanson, Melvyn B. |
| contents | The $h$-fold sumset of a set $A$ of integers is the set of all sums of $h$ not necessarily distinct elements of $A$. Let $(A_q)_{q=1}^{\infty}$ be a strictly decreasing sequence of sets of integers and let $A = \bigcap_{q=1}^{\infty} A_q$. Then $hA \subseteq \bigcap_{q=1}^{\infty} hA_q$ for all $h \geq 1$. Let $\mathcal{H}(A_q) = \{h \geq 1: hA = \bigcap_{q=1}^{\infty} hA_q\}$. The arithmetical structure of the sets $\mathcal{H}(A_q)$ is unknown. It is proved that for every $h_0 \geq 2$ there exist sequences $(A_q)_{q=1}^{\infty}$ such that $\{1,\ldots, h_0-1\} \subseteq \mathcal{H}(A_q)$ but $h_0 \notin \mathcal{H}(A_q)$ and also that there exist sequences $(A_q)_{q=1}^{\infty}$ such that $\{1, h_0 \} \subseteq \mathcal{H}(A_q)$ but $\{2,3, \ldots, h_0-1\} \cap \mathcal{H}(A_q) = \emptyset$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14510 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Arithmetical structure of sumset intersections Marques, Diego Nathanson, Melvyn B. Number Theory 11B13, 11B05, 11B75, 11P70 The $h$-fold sumset of a set $A$ of integers is the set of all sums of $h$ not necessarily distinct elements of $A$. Let $(A_q)_{q=1}^{\infty}$ be a strictly decreasing sequence of sets of integers and let $A = \bigcap_{q=1}^{\infty} A_q$. Then $hA \subseteq \bigcap_{q=1}^{\infty} hA_q$ for all $h \geq 1$. Let $\mathcal{H}(A_q) = \{h \geq 1: hA = \bigcap_{q=1}^{\infty} hA_q\}$. The arithmetical structure of the sets $\mathcal{H}(A_q)$ is unknown. It is proved that for every $h_0 \geq 2$ there exist sequences $(A_q)_{q=1}^{\infty}$ such that $\{1,\ldots, h_0-1\} \subseteq \mathcal{H}(A_q)$ but $h_0 \notin \mathcal{H}(A_q)$ and also that there exist sequences $(A_q)_{q=1}^{\infty}$ such that $\{1, h_0 \} \subseteq \mathcal{H}(A_q)$ but $\{2,3, \ldots, h_0-1\} \cap \mathcal{H}(A_q) = \emptyset$. |
| title | Arithmetical structure of sumset intersections |
| topic | Number Theory 11B13, 11B05, 11B75, 11P70 |
| url | https://arxiv.org/abs/2603.14510 |