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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2501.11736 |
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| _version_ | 1866916574690017280 |
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| author | Schmutz, Eric Tait, Michael |
| author_facet | Schmutz, Eric Tait, Michael |
| contents | Let $η_{g}(n) $ be the smallest cardinality that $A\subseteq {\mathbb Z}$ can have if $A$ is a $g$-difference basis for $[n]$ (i.e, if, for each $x\in [n]$, there are {\em at least} $g$ solutions to $a_{1}-a_{2}=x$ ). We prove that the finite, non-zero limit $\lim\limits_{n\rightarrow \infty}\frac{η_{g}(n)}{\sqrt{n}}$ exists, answering a question of Kravitz. We also investigate a similar problem in the setting of a vector space over a finite field.
Let $α_g(n)$ be the largest cardinality that $A\subseteq [n]$ can have if, for all nonzero $x$, $a_{1}-a_{2}=x$ has {\em at most} $g$ solutions. We also prove that $α_g(n)={\sqrt{gn}}(1+o_{g}(1))$ as $n\rightarrow\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_11736 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cardinalities of $g$-difference sets Schmutz, Eric Tait, Michael Combinatorics 05B10, 11B13, 11B75 Let $η_{g}(n) $ be the smallest cardinality that $A\subseteq {\mathbb Z}$ can have if $A$ is a $g$-difference basis for $[n]$ (i.e, if, for each $x\in [n]$, there are {\em at least} $g$ solutions to $a_{1}-a_{2}=x$ ). We prove that the finite, non-zero limit $\lim\limits_{n\rightarrow \infty}\frac{η_{g}(n)}{\sqrt{n}}$ exists, answering a question of Kravitz. We also investigate a similar problem in the setting of a vector space over a finite field. Let $α_g(n)$ be the largest cardinality that $A\subseteq [n]$ can have if, for all nonzero $x$, $a_{1}-a_{2}=x$ has {\em at most} $g$ solutions. We also prove that $α_g(n)={\sqrt{gn}}(1+o_{g}(1))$ as $n\rightarrow\infty$. |
| title | Cardinalities of $g$-difference sets |
| topic | Combinatorics 05B10, 11B13, 11B75 |
| url | https://arxiv.org/abs/2501.11736 |