Pollard's theorem in general abelian groups
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arXiv
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| Format: | Preprint |
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2026
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| author | Grynkiewicz, David J. Wang, Runze |
| author_facet | Grynkiewicz, David J. Wang, Runze |
| contents | We make further progress towards a Kneser-type generalization of Pollard's Theorem to general abelian groups. For two sets $A$ and $B$ in an abelian group $G$, the \emph{$t$-popular sumset} of $A$ and $B$, denoted by $A+_t B$, is the set of elements in $G$ each with at least $t$ representations of the form $a+b$, where $a\in A$ and $b\in B$. For $|A|,\, |B|\ge t\geq 2$, we prove that if \begin{align*} \sum_{i=1}^t |A+_i B|< t|A|+t|B|-\frac{4}{3}t^2+\frac{2}{3}t, \end{align*} then there exist $A'\subseteq A$ and $B'\subseteq B$ with $|A\setminus A'|+|B\setminus B'|\le t-1$, $A'+_t B'=A'+B'=A+_t B$, and $ \sum_{i=1}^t |A+_i B|\ge t|A|+t|B|-t|H|,$ where $H$ is the stabilizer of $A'+B'=A+_t B$. Our result improves the main quadratic term in the previous best bound from $-2t^2$ to $-\frac{4}{3}t^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_17922 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Pollard's theorem in general abelian groups Grynkiewicz, David J. Wang, Runze Number Theory 11P70, 11B13 We make further progress towards a Kneser-type generalization of Pollard's Theorem to general abelian groups. For two sets $A$ and $B$ in an abelian group $G$, the \emph{$t$-popular sumset} of $A$ and $B$, denoted by $A+_t B$, is the set of elements in $G$ each with at least $t$ representations of the form $a+b$, where $a\in A$ and $b\in B$. For $|A|,\, |B|\ge t\geq 2$, we prove that if \begin{align*} \sum_{i=1}^t |A+_i B|< t|A|+t|B|-\frac{4}{3}t^2+\frac{2}{3}t, \end{align*} then there exist $A'\subseteq A$ and $B'\subseteq B$ with $|A\setminus A'|+|B\setminus B'|\le t-1$, $A'+_t B'=A'+B'=A+_t B$, and $ \sum_{i=1}^t |A+_i B|\ge t|A|+t|B|-t|H|,$ where $H$ is the stabilizer of $A'+B'=A+_t B$. Our result improves the main quadratic term in the previous best bound from $-2t^2$ to $-\frac{4}{3}t^2$. |
| title | Pollard's theorem in general abelian groups |
| topic | Number Theory 11P70, 11B13 |
| url | https://arxiv.org/abs/2601.17922 |