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arXiv · 2406.03312

On the sumsets of exceptional units in quaternion rings

Hassan Cheraghpour·David Dolžan
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Abstract

We investigate sums of exceptional units in a quaternion ring $H(R)$ over a finite commutative ring $R$. We prove that in order to find the number of representations of an element in $H(R)$ as a sum of $k$ exceptional units for some integer $k \geq 2$, we can limit ourselves to studying the quaternion rings over local rings. For a local ring $R$ of even order, we find the number of representations of an element of $H(R)$ as a sum of $k$ exceptional units for any integer $k \geq 2$. For a local ring $R$ of odd order, we find either the number or the bounds for the number of representations of an element of $H(R)$ as a sum of $2$ exceptional units.

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Hassan Cheraghpour, David Dolžan. 2024-06-05. On the sumsets of exceptional units in quaternion rings. https://arxiv.org/abs/2406.03312

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