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.