Some Refined Results on the Mixed Littlewood Conjecture for Pseudo-Absolute Values

Abstract

In this paper, we study the mixed Littlewood conjecture with pseudo-absolute values. For any pseudo absolute value sequence D, we obtain the sharp criterion such that for almost every α the inequalityhas infinitely many coprime solutions (n, p) ∈ N × Z for a certain one-parameter family of ψ. Also under minor condition on pseudo absolute value sequences D 1 ,D 2 , · · · , D k , we obtain a sharp criterion on general sequence ψ(n) such that for almost every α the inequalityhas infinitely many coprime solutions (n, p) ∈ N × Z.

Badly approximable points for diagonal approximation in solenoids

“…There are several natural ways of extending the classical study of Diophantine approximation to spaces of p-adic numbers. The first and most obvious, which was pioneered by K. Mahler [24], Jarník [16], and Lutz [23], is to study approximations in Q p in the usual sense by simply restricting to Z p (see also [2,3,12,14,21,22,27]). A second way, which was introduced by Choi and Vaaler [7] and studied using techniques from the geometry of numbers over the adeles [4,5], is to work in projective space over Q p (see also [10,11]).…”