One step further of an inverse theorem for the restricted set addition in $\mathbb{Z}/p\mathbb{Z}$
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| Format: | Preprint |
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2024
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| _version_ | 1866912089536200704 |
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| author | Urbano, David Fernando Daza González-Martínez, René Mason, Mario Huicochea Cantoral, Amanda Montejano |
| author_facet | Urbano, David Fernando Daza González-Martínez, René Mason, Mario Huicochea Cantoral, Amanda Montejano |
| contents | Let $A$ and $B$ be sets of $k\ge5$ elements in $F=\mathbb{Z}/p\mathbb{Z}$ the field with $p>2k-2$ elements. We denote by $A\dot{+}B$ the set of different elements of $F$ that can be written in the form $a+b$, where $a\in A$, $b\in B$, $a\neq b$. The number of elements of this set is at least $2k-3$. Károlyi showed that, except from some particular cases, The equality can only occur if $A = B$ and $A$ is an arithmetic progression with non zero difference. We prove that in the case that $|A\dot{+}B| = 2k - 2$ and $|A|=|B|$ the equality $A=B$ holds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20767 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | One step further of an inverse theorem for the restricted set addition in $\mathbb{Z}/p\mathbb{Z}$ Urbano, David Fernando Daza González-Martínez, René Mason, Mario Huicochea Cantoral, Amanda Montejano Number Theory 11P70 Let $A$ and $B$ be sets of $k\ge5$ elements in $F=\mathbb{Z}/p\mathbb{Z}$ the field with $p>2k-2$ elements. We denote by $A\dot{+}B$ the set of different elements of $F$ that can be written in the form $a+b$, where $a\in A$, $b\in B$, $a\neq b$. The number of elements of this set is at least $2k-3$. Károlyi showed that, except from some particular cases, The equality can only occur if $A = B$ and $A$ is an arithmetic progression with non zero difference. We prove that in the case that $|A\dot{+}B| = 2k - 2$ and $|A|=|B|$ the equality $A=B$ holds. |
| title | One step further of an inverse theorem for the restricted set addition in $\mathbb{Z}/p\mathbb{Z}$ |
| topic | Number Theory 11P70 |
| url | https://arxiv.org/abs/2410.20767 |