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p-adic quotient sets: cubic forms

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Abstract

For \(A\subseteq \{1, 2, \ldots \}\), we consider \(R(A)=\{a/b: a, b\in A\}\). It is an open problem to study the denseness of R(A) in the p-adic numbers when A is the set of nonzero values assumed by a cubic form. We study this problem for the cubic forms \(ax^3+by^3\), where a and b are integers. We also prove that if A is the set of nonzero values assumed by a non-degenerate, integral, and primitive cubic form with more than 9 variables, then R(A) is dense in \({\mathbb {Q}}_p\).

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Acknowledgements

The authors are grateful to Piotr Miska for pointing out an error in an earlier draft of the article and for many helpful discussions.

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  1. Department of Mathematics, Indian Institute of Technology Guwahati, Assam, PIN-781039, India

    Deepa Antony & Rupam Barman

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  1. Deepa Antony
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  2. Rupam Barman
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Correspondence to Rupam Barman.

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Antony, D., Barman, R. p-adic quotient sets: cubic forms. Arch. Math. 118, 143–149 (2022). https://doi.org/10.1007/s00013-021-01689-8

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