2009/06/05 by Felix Breuer, Breuer, Felix, Frederik von Heymann +1
Mathematics · #05A15 #11H06 #11P21 #11Y55 #52C05 #Advanced Differential Equations and Dynamical Systems #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05A15 #msc:11H06 #msc:11P21 #msc:11Y55 #msc:52C05
paper · pdf · doi:10.48550/arxiv.0906.1191
32 pages, 11 figures
arxiv created 2009/06/05 · openalex publication_date 2009/06/05 · arxiv updated 2009/12/01 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
A staircase is the set of points in Z2 below a given rational line in the plane that have Manhattan Distance less than 1 to the line. Staircases are closely related to Beatty and Sturmian sequences of rational numbers. Connecting the geometry and the number theoretic concepts, we obtain three equivalent characterizations of Sturmian sequences of rational numbers, as well as a new proof of Barvinok's Theorem in dimension two, a recursion formula for Dedekind-Carlitz polynomials and a partially new proof of White's characterization of empty lattice tetrahedra. Our main tool is a recursive description of staircases in the spirit of the Euclidean Algorithm.