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arXiv · 2006.02429

Infinite co-minimal pairs involving lacunary sequences and generalisations to higher dimensions

Arindam Biswas·Jyoti Prakash Saha
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Abstract

The study of minimal complements in a group or a semigroup was initiated by Nathanson. The notion of minimal complements and being a minimal complement leads to the notion of co-minimal pairs which was considered in a prior work of the authors. In this article, we study which type of subsets in the integers and free abelian groups of higher rank can be a part of a co-minimal pair. We show that a majority of lacunary sequences have this property. From the conditions established, one can show that any infinite subset of any finitely generated abelian group has uncountably many subsets which is a part of a co-minimal pair. Further, the uncountable collection of sets can be chosen so that they satisfy certain algebraic properties.

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Arindam Biswas, Jyoti Prakash Saha. 2020-06-03. Infinite co-minimal pairs involving lacunary sequences and generalisations to higher dimensions. https://doi.org/10.1007/s11139-021-00442-7

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