2 research outputs found
Asymptotic diophantine approximation:the multiplicative case
Let and be irrational real numbers and 0<\F<1/30. We prove
a precise estimate for the number of positive integers that satisfy
\|q\alpha\|\cdot\|q\beta\|<\F. If we choose \F as a function of we get
asymptotics as gets large, provided \F Q grows quickly enough in terms of
the (multiplicative) Diophantine type of , e.g., if
is a counterexample to Littlewood's conjecture then we only
need that \F Q tends to infinity. Our result yields a new upper bound on sums
of reciprocals of products of fractional parts, and sheds some light on a
recent question of L\^{e} and Vaaler.Comment: To appear in Ramanujan Journa