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Preprint (Author's original)arXiv.org - Non-exclusive license to distribute, Open
Preprint
Cardinalities of $g$-difference sets
20 Jan 2025
Abstract
Let $\eta_{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 $\lim\limits_{n\rightarrow
\infty}\frac{\eta_{g}(n)}{\sqrt{n}}$ exists, answering a question of Kravitz.
We also investigate a similar problem in the setting of a vector space over a
finite field.
Let $\alpha_g(n)$ 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 $\alpha_g(n)={\sqrt{gn}}(1+o_{g}(1))$ as $n\rightarrow\infty$.
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Details
- Title
- Cardinalities of $g$-difference sets
- Creators
- Eric SchmutzMichael Tait
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991022020437804721