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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.12319 |
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| _version_ | 1866909780033929216 |
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| author | Wang, Jing |
| author_facet | Wang, Jing |
| contents | Let $G$ be a finite abelian group of order $n$, and for each $a\in G$ and integer $1\le h\le n$ let $\mathcal{F}_a(h)$ denote the family of all $h$-element subsets of $G$ whose sum is $a$. A problem posed by Katona and Makar-Limanov is to determine whether the minimum and maximum sizes of the families $\mathcal{F}_a(h)$ (as $a$ ranges over $G$) become asymptotically equal as $n\rightarrow \infty$ when $h=\left\lfloor\frac{n}{2}\right\rfloor$. We affirmatively answer this question and in fact show that the same asymptotic equality holds for every $4\leq h\leq \left\lfloor\frac{n}{2}\right\rfloor+1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12319 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The asymptotic uniform distribution of subset sums Wang, Jing Combinatorics Let $G$ be a finite abelian group of order $n$, and for each $a\in G$ and integer $1\le h\le n$ let $\mathcal{F}_a(h)$ denote the family of all $h$-element subsets of $G$ whose sum is $a$. A problem posed by Katona and Makar-Limanov is to determine whether the minimum and maximum sizes of the families $\mathcal{F}_a(h)$ (as $a$ ranges over $G$) become asymptotically equal as $n\rightarrow \infty$ when $h=\left\lfloor\frac{n}{2}\right\rfloor$. We affirmatively answer this question and in fact show that the same asymptotic equality holds for every $4\leq h\leq \left\lfloor\frac{n}{2}\right\rfloor+1$. |
| title | The asymptotic uniform distribution of subset sums |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2505.12319 |