Diophantine Analysis by Jörn Steuding

Author:Jörn Steuding
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

2.White picks a closed interval of length .

3.Black picks a closed interval of length .

4.And so on...

By Banach’s fixpoint theorem, consists of a single point, x say. If , White wins the game. Otherwise, Black wins.

By requiring that the initial ball is chosen in a particular way, we can use the above for bounded sets without breaking the game.

We are concerned with winning strategies for the game. In particular, we will prove that if White has a winning strategy, then the set F is large.

Definition 5.10

A set F is said to be -winning if White can always win the Schmidt game with these parameters. A set F is said to be -winning if it is -winning for any .

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