A Note on Weighted Badly Approximable Linear Forms

Abstract

Abstract. We prove a result in the area of twisted Diophantine approximation related to the theory of Schmidt games. In particular, under certain restrictions we give a affirmative answer to the analogue in this setting of a famous conjecture of Schmidt from Diophantine approximation.

Sets of inhomogeneous linear forms can be not isotropically winning

“…Harrap and Moshchevitin in [7] showed that this set is winning provided that θ θ θ ∈ Bad(k, n, m). In [2] it was proved that if we suppose that θ θ θ ∈ Bad(k, n, m) the set Bad θ θ θ (k, n, m) is isotropically winning 1 .…”

“…We define a vector e e e ν from the conditions (7) e e e ν ∈ π ν , |e e e ν | = 1, (e e e ν , z z z ν ) = 0 so e e e ν is parallel to π ν and orthogonal to z z z ν . Define the rectangle…”

Diophantine transference inequalities: weighted, inhomogeneous, and intermediate exponents

“…Then there exists ε = ε(A) > 0 such that dim H Bad ε r,s ( t A) = n. This is often referred to as twisted diophantine approximation: the inhomogeneous shift is metric. The analogous problem for weighted badly approximable matrices has hitherto been investigated in [20,6]; therein the object of study is Bad r,s ( t A) := ∪ ε>0 Bad ε r,s ( t A). Our conclusion is stronger than the assertion that dim H Bad r,s ( t A) = n.…”

Badly approximable points in twisted Diophantine approximation and Hausdorff dimension

“…. , j n ), Harrap and Moshchevitin have extended to weighted linear forms in higher dimension and improved to winning the result in [10] (see [11]).…”