2016/07/14 by Ofir David, Uri Shapira
Mathematics · #Algebraic Geometry and Number Theory #Closure (psychology) #Conjecture #Diagonal #Dirichlet distribution #Geometric and Algebraic Topology #Group (periodic table) #Logarithm #Mathematical Dynamics and Fractals #Unit (ring theory) #Unit square #math.DS #math.NT
paper · pdf · doi:10.1093/imrn/rnw324
published in International Mathematics Research Notices, rnw324 (Oxford University Press) · 29 pages, 2 figures
arxiv created 2016/07/14 · openalex created_date 2016/08/23 · openalex publication_date 2017/01/19 · arxiv updated 2017/07/04 · openalex updated_date 2026/08/05
We study the collection of points on the modular surface obtained from the logarithm embeddings of the groups of units in totally real cubic number fields (which we term Dirichlet shapes of unit lattices). We conjecture that this set is dense and show that its closure contains countably many explicit curves and give a strategy to show that it has non-empty interior. The results are obtained by constructing explicit families of orders (generalizing the so called “simplest cubic fields”) and calculating their groups of units. We also address the question of escape of mass for the compact orbits of the diagonal group associated to these orders.