How Many Reflections Make a Dihedral Set Large?
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866911540578353152 |
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| author | Greenfeld, Be'eri King, George Li, Xiaoxuan Tacheny, Sam |
| author_facet | Greenfeld, Be'eri King, George Li, Xiaoxuan Tacheny, Sam |
| contents | Given a size-$k$ subset $S$ of a group $G$, how large can the product set $S^n$ be? We study this question, at several layers of refinement, for the infinite dihedral group.
First, we give an explicit formula for the maximum size of $S^n$ among all size-$k$ subsets with a prescribed number of reflections. We then determine the optimal number of reflections that a size-$k$ set should contain in order to maximize $|S^n|$.
When $k$ is fixed and $n\to\infty$, we obtain a clean asymptotic expression for the maximal size of $S^n$. Moreover, we compute this asymptotic separately for each fixed number of reflections in $S$. We show that the number of reflections influences the asymptotic size of $S^n$ only through a multiplicative coefficient, which admits a direct probabilistic interpretation.
Finally, we compute the growth exponent of the maximum of $|S^n|$ when~$k=~n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22533 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | How Many Reflections Make a Dihedral Set Large? Greenfeld, Be'eri King, George Li, Xiaoxuan Tacheny, Sam Group Theory Combinatorics Given a size-$k$ subset $S$ of a group $G$, how large can the product set $S^n$ be? We study this question, at several layers of refinement, for the infinite dihedral group. First, we give an explicit formula for the maximum size of $S^n$ among all size-$k$ subsets with a prescribed number of reflections. We then determine the optimal number of reflections that a size-$k$ set should contain in order to maximize $|S^n|$. When $k$ is fixed and $n\to\infty$, we obtain a clean asymptotic expression for the maximal size of $S^n$. Moreover, we compute this asymptotic separately for each fixed number of reflections in $S$. We show that the number of reflections influences the asymptotic size of $S^n$ only through a multiplicative coefficient, which admits a direct probabilistic interpretation. Finally, we compute the growth exponent of the maximum of $|S^n|$ when~$k=~n$. |
| title | How Many Reflections Make a Dihedral Set Large? |
| topic | Group Theory Combinatorics |
| url | https://arxiv.org/abs/2603.22533 |