Research output: Working paper

**On multiplicatively badly approximable numbers.** / Badziahin, Dzmitry.

Research output: Working paper

Badziahin, D 2011 'On multiplicatively badly approximable numbers'. <http://arxiv.org/abs/1101.1855>

Badziahin, D. (2011). *On multiplicatively badly approximable numbers*. http://arxiv.org/abs/1101.1855

Badziahin D. On multiplicatively badly approximable numbers. 2011 Jan.

@techreport{95b18c8f628b4a698ad80be6a9b21653,

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. ",

author = "Dzmitry Badziahin",

year = "2011",

month = jan,

language = "English",

type = "WorkingPaper",

}

TY - UNPB

T1 - On multiplicatively badly approximable numbers

AU - Badziahin, Dzmitry

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 -