## On multiplicatively badly approximable numbers

Research output: Working paper

### Standard

2011.

Research output: Working paper

### Author

title = "On multiplicatively badly approximable numbers",
abstract = "The Littlewood Conjecture states that liminf_{q\to \infty} q . ||qx|| . ||qy|| = 0 for all pairs (x,y) of real numbers. We show that with the additional factor of log q . loglog q the statement is false. Indeed, our main result implies that the set of (x,y) for which liminf_{q\to\infty} q . log q . loglog q . ||qx|| . ||qy|| > 0 is of full dimension. ",
year = "2011",
month = jan,
language = "English",
type = "WorkingPaper",

}

TY - UNPB

T1 - On multiplicatively badly approximable numbers

PY - 2011/1

Y1 - 2011/1

N2 - The Littlewood Conjecture states that liminf_{q\to \infty} q . ||qx|| . ||qy|| = 0 for all pairs (x,y) of real numbers. We show that with the additional factor of log q . loglog q the statement is false. Indeed, our main result implies that the set of (x,y) for which liminf_{q\to\infty} q . log q . loglog q . ||qx|| . ||qy|| > 0 is of full dimension.

AB - The Littlewood Conjecture states that liminf_{q\to \infty} q . ||qx|| . ||qy|| = 0 for all pairs (x,y) of real numbers. We show that with the additional factor of log q . loglog q the statement is false. Indeed, our main result implies that the set of (x,y) for which liminf_{q\to\infty} q . log q . loglog q . ||qx|| . ||qy|| > 0 is of full dimension.

M3 - Working paper

BT - On multiplicatively badly approximable numbers

ER -