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Pré-Publication, Document De Travail Année : 2023

On the connectedness of arithmetic hyperplanes

Résumé

We are interested in a fundamental structure of discrete geometry, the arithmetic hyper- planes, and more particularly in their connectedness. Many works have studied a connectedness defined from the neighbourhood by faces and have observed a percolation phenomenon. We propose an exten- sion of these results in the case of connectivity defined from general neighbourhoods. Beyond the new concepts required by this extension, the main contribution of the article lies in the use of analysis to solve this arithmetic problem. The study of the connecting thickness as a function then brings out discontinuities at each rational point but an underlying fractal that is much more regular in the irrational case. Thus, the consideration of irrational vectors allows a simpler approach to the connectedness of arithmetic hyperplanes.
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Dates et versions

hal-04310348 , version 1 (27-11-2023)

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  • HAL Id : hal-04310348 , version 1

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Bastien Laboureix, Éric Domenjoud. On the connectedness of arithmetic hyperplanes. 2023. ⟨hal-04310348⟩
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