# PhD position: Coding of arithmetic discrete hyperplanes and numeration systems

Team : ADAGIO

Supervisors : Isabelle Debled-Rennesson et Eric Domenjoud

Coding of arithmetic discrete hyperplanes and numeration systems

An arithmetic discrete hyperplane H in ℤⁿ is the set of integer points between two parallel real hyperplanes. Formally, it is the set H(v,μ,θ) = {x ∈ ℤⁿ | 0 ≤ ⟨v,x⟩+μ < θ} where ⟨.,.⟩ is the usual scalar product in ℝⁿ, v ∈ ℝⁿ\{0} is the normal vector, μ ∈ ℝ is the shift and θ ∈ ℝ is the thickness. Topological properties, especially conectedness, of these objects have been widely studied. The notoin of connectedness is intimately linked to the notion of neighbourhood of a point. The study of conectedness in the particular case where the neighbourhood in ℤⁿ of a point x is defined as the set {y ∈ ℤⁿ | ∥x-y∥₁ ≤ 1} allowed to completely characterize of the cases where H(v,μ,θ) is connected according to v, μ and θ. A particular set of normal vectors arose for which H(v,μ,θ) has very specific properties. In this case, one can define a numeration system, called Δ-numeration, which is attached to the normal vector v.

The goal of the thesis is on one hand to generalize the results about conectedness of discrete hyperplanes when connectedness is defined by an arbitrary notion of neighbourhood of a point, and on the other hand to study and formalize Δ-numeration in this general framework.