Badly approximable points on planar curves and winning

TY - JOUR

T1 - Badly approximable points on planar curves and winning

AU - An, Jinpeng

AU - Beresnevich, Victor

AU - Velani, Sanju

N1 - © 2017 The Author(s).

PY - 2018/1/14

Y1 - 2018/1/14

N2 - For any i,j>0 with i+j=1, let Bad(i,j) denote the set of points (x,y)∈R 2 such that max⁡{‖qx‖ 1/i,‖qy‖ 1/j}>c/q for some positive constant c=c(x,y) and all q∈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 generalisations 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.

AB - For any i,j>0 with i+j=1, let Bad(i,j) denote the set of points (x,y)∈R 2 such that max⁡{‖qx‖ 1/i,‖qy‖ 1/j}>c/q for some positive constant c=c(x,y) and all q∈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 generalisations 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.

KW - Inhomogeneous Diophantine approximation

KW - Non-degenerate curves

KW - Schmidt games

UR - http://www.scopus.com/inward/record.url?scp=85034573471&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2017.11.009

DO - 10.1016/j.aim.2017.11.009

M3 - Article

SN - 0001-8708

VL - 324

SP - 148

EP - 202

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -