Projects per year
Abstract
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 nondegenerate 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.
Original language  English 

Pages (fromto)  148–202 
Number of pages  55 
Journal  Advances in Mathematics 
Volume  324 
Early online date  21 Nov 2017 
DOIs  
Publication status  Published  14 Jan 2018 
Bibliographical note
© 2017 The Author(s).Keywords
 Inhomogeneous Diophantine approximation
 Nondegenerate curves
 Schmidt games
Profiles
Projects
 1 Finished

Programme GrantNew Frameworks in metric Number Theory
1/06/12 → 30/11/18
Project: Research project (funded) › Research