Bohr sets and multiplicative diophantine approximation

TY - JOUR

T1 - Bohr sets and multiplicative diophantine approximation

AU - Chow, Samuel Khai Ho

N1 - This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details.

PY - 2018/6/15

Y1 - 2018/6/15

N2 - In two dimensions, Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fibre version of Gallagher's theorem, sharpening and making unconditional a result recently obtained conditionally by Beresnevich, Haynes and Velani. The idea is to find large generalised arithmetic progressions within inhomogeneous Bohr sets, extending a construction given by Tao. This precise structure enables us to verify the hypotheses of the Duffin--Schaeffer theorem for the problem at hand, via the geometry of numbers.

AB - In two dimensions, Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fibre version of Gallagher's theorem, sharpening and making unconditional a result recently obtained conditionally by Beresnevich, Haynes and Velani. The idea is to find large generalised arithmetic progressions within inhomogeneous Bohr sets, extending a construction given by Tao. This precise structure enables us to verify the hypotheses of the Duffin--Schaeffer theorem for the problem at hand, via the geometry of numbers.

KW - metric diophantine approximation

KW - additive combinatorics

KW - geometry of numbers

UR - http://www.scopus.com/inward/record.url?scp=85048554098&partnerID=8YFLogxK

U2 - 10.1215/00127094-2018-0001

DO - 10.1215/00127094-2018-0001

M3 - Article

SN - 0012-7094

VL - 167

SP - 1623

EP - 1642

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

IS - 9

ER -