A randomized version of the Littlewood Conjecture

TY - UNPB

T1 - A randomized version of the Littlewood Conjecture

AU - Haynes, Alan

AU - Koivusalo, Henna

N1 - 6 pages, newer version: added reference [1]

PY - 2016/10/25

Y1 - 2016/10/25

N2 - The Littlewood Conjecture in Diophantine approximation can be thought of as a problem about covering the plane by a union of hyperbolas centered at rational points. In this paper we consider the problem of translating the center of each hyperbola by a random amount which depends on the denominator of the corresponding rational. Using a randomized covering argument we prove that, not only is this randomized version of the Littlewood Conjecture true for almost all choices of centers, an even stronger statement with an extra factor of a logarithm also holds.

AB - The Littlewood Conjecture in Diophantine approximation can be thought of as a problem about covering the plane by a union of hyperbolas centered at rational points. In this paper we consider the problem of translating the center of each hyperbola by a random amount which depends on the denominator of the corresponding rational. Using a randomized covering argument we prove that, not only is this randomized version of the Littlewood Conjecture true for almost all choices of centers, an even stronger statement with an extra factor of a logarithm also holds.

KW - math.NT

KW - 11J13, 60D05

M3 - Working paper

BT - A randomized version of the Littlewood Conjecture

ER -