Sets of bounded remainder for the continuous irrational rotation on [0,1)²

2016/03/01 by Sigrid Grepstad, Grepstad, Sigrid, Gerhard Larcher +1
Mathematics · #11J71 #11K38 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematics and Applications #Number Theory (math.NT) #Point processes and geometric inequalities #math.NT #msc:11J71 #msc:11K38

paper · pdf · doi:10.48550/arxiv.1603.00207

arxiv created 2016/03/01 · openalex publication_date 2016/03/01 · arxiv updated 2016/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study sets of bounded remainder for the two-dimensional continuous irrational rotation (\x₁+t\, \x₂+tα\)_{t ≥ 0} in the unit square. In particular, we show that for almost all α and every starting point (x₁, x₂), every polygon S with no edge of slope α is a set of bounded remainder. Moreover, every convex set S whose boundary is twice continuously differentiable with positive curvature at every point is a bounded remainder set for almost all α and every starting point (x₁, x₂). Finally we show that these assertions are, in some sense, best possible.

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