A Note on Badly Approximable Linear Forms on Manifolds

Abstract

This paper is motivated by Davenport's problem and the subsequent work regarding badly approximable points in submanifolds of a Euclidian space. We study the problem in the area of twisted Diophantine approximation and present two different approaches. The first approach shows that, under a certain restriction, any countable intersection of the sets of weighted badly approximable points on any non-degenerate C 1 submanifold of R n has full dimension. In the second approach we introduce the property of isotropically winning and show that the sets of weighted badly approximable points are isotropically winning under the same restriction as above.

Sets of inhomogeneous linear forms can be not isotropically winning

“…where || • || stands for the distance to the nearest integer is badly approximable. We prove a statement complementary to our recent result from [2]. We construct θ θ θ such that the set Bad θ θ θ := {(η 1 , η 2 ) : inf…”

“…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 .…”

“…Remark 1. From the result of the paper [2] it follows that the vector θ θ θ constructed in Theorem 3.1 does not belong to the set…”

“…There are different modifications of Schmidt's game: strong game and absolute game introduced in [15], hyperplane absolute game introduced in [14], potential game considered in [6] and some others. In our recent paper [2] we introduced isotropically winning sets. Let us describe here some of these generalizations in more details.…”

“…In fact the approach from[2] gives a little bit more. Instead of property that for any subspace A the intersection E ∩ A is 1/2 winning in A one can see that it is α-winning for all α ∈ (0, 1/2].…”