2014/08/29 by Jinpeng An, Victor Beresnevich, An, Jinpeng +3 · 2 citations
Mathematics · #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1409.0064
openalex publication_date 2014/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For any i,j>0 with i+j =1, let Bad(i,j) denote the set of points (x,y) ∈ R2 such that max \ ||qx||1/i, ||qy||1/j \ > c/q for some positive constant c = c(x,y) and all q in N. We show that \Bad(i,j) ∩ C is winning in the sense of Schmidt games for a large class of planar curves C, namely, everywhere non-degenerate planar curves and straight lines satisfying a natural Diophantine condition. This strengthens recent results solving a problem of Davenport from the sixties. In short, within the context of Davenport's problem, the winning statement is best possible. Furthermore, we obtain the inhomogeneous generalizations of the winning results for planar curves and lines and also show that the inhomogeneous form of Bad(i,j) is winning for two dimensional Schmidt games.