The mixed Schmidt conjecture in the theory of Diophantine approximation

TY - JOUR

T1 - The mixed Schmidt conjecture in the theory of Diophantine approximation

AU - Badziahin, Dzmitry

AU - Levesley, Jason

AU - Velani, Sanju

PY - 2011/7

Y1 - 2011/7

N2 - Let $\mathcal{D}=(d_n)_{n=1}^\infty$ be a bounded sequence of integers with $d_n\ge 2$ and let $(i, j)$ be a pair of strictly positive numbers with $i+j=1$. We prove that the set of $x \in \RR$ for which there exists some constant $c(x) > 0$ such that \[ \max\{|q|_\DDD^{1/i}, \|qx\|^{1/j}\} > c(x)/ q \qquad \forall q \in \NN \] is one quarter winning (in the sense of Schmidt games). Thus the intersection of any countable number of such sets is of full dimension. In turn, this establishes the natural analogue of Schmidt's conjecture within the framework of the de Mathan-Teuli\'e conjecture -- also known as the `Mixed Littlewood Conjecture'.

AB - Let $\mathcal{D}=(d_n)_{n=1}^\infty$ be a bounded sequence of integers with $d_n\ge 2$ and let $(i, j)$ be a pair of strictly positive numbers with $i+j=1$. We prove that the set of $x \in \RR$ for which there exists some constant $c(x) > 0$ such that \[ \max\{|q|_\DDD^{1/i}, \|qx\|^{1/j}\} > c(x)/ q \qquad \forall q \in \NN \] is one quarter winning (in the sense of Schmidt games). Thus the intersection of any countable number of such sets is of full dimension. In turn, this establishes the natural analogue of Schmidt's conjecture within the framework of the de Mathan-Teuli\'e conjecture -- also known as the `Mixed Littlewood Conjecture'.

KW - Number Theory

UR - http://www.scopus.com/inward/record.url?scp=80053081789&partnerID=8YFLogxK

U2 - 10.1112/S0025579311002075

DO - 10.1112/S0025579311002075

M3 - Article

SN - 0025-5793

VL - 57

SP - 239

EP - 245

JO - Mathematika

JF - Mathematika

IS - 2

ER -