Possible Sizes of Sumsets
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911255249289216 |
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| author | Rajagopal, Isaac |
| author_facet | Rajagopal, Isaac |
| contents | Nathanson introduced the range of cardinalities of $h$-fold sumsets $R(h,k) := \{|hA|:A \subset \mathbb{Z} \text{ and }|A| = k\}.$ Following a remark of Erdős and Szemerédi that determined the form of $R(h,k)$ when $h=2$, Nathanson asked what the form of $R(h,k)$ is for arbitrary $h, k \in \mathbb{N}$. For $h \in \mathbb{N}$, we prove there is some constant $k_h \in \mathbb{N}$ such that if $k > k_h$, then $R(h,k)$ is the entire interval $\left[hk-h+1,\binom{h+k-1}{h}\right]$ except for a specified set of $\binom{h-1}{2}$ numbers. Moreover, we show that one can take $k_3 = 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23022 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Possible Sizes of Sumsets Rajagopal, Isaac Combinatorics Number Theory 11P70 Nathanson introduced the range of cardinalities of $h$-fold sumsets $R(h,k) := \{|hA|:A \subset \mathbb{Z} \text{ and }|A| = k\}.$ Following a remark of Erdős and Szemerédi that determined the form of $R(h,k)$ when $h=2$, Nathanson asked what the form of $R(h,k)$ is for arbitrary $h, k \in \mathbb{N}$. For $h \in \mathbb{N}$, we prove there is some constant $k_h \in \mathbb{N}$ such that if $k > k_h$, then $R(h,k)$ is the entire interval $\left[hk-h+1,\binom{h+k-1}{h}\right]$ except for a specified set of $\binom{h-1}{2}$ numbers. Moreover, we show that one can take $k_3 = 2$. |
| title | Possible Sizes of Sumsets |
| topic | Combinatorics Number Theory 11P70 |
| url | https://arxiv.org/abs/2510.23022 |