Twisted inhomogeneous Diophantine approximation and badly approximable sets

Abstract

For any real pair i, j ≥ 0 with i + j = 1 let Bad(i, j) denote the set of (i, j)-badly approximable pairs. That is, Bad(i, j) consists of irrational vectors x := (x 1 , x 2 ) ∈ R 2 for which there exists a positive constant c(x) such that max qx 1 1/i , qx 2 1/j > c(x)/q ∀ q ∈ N.A new characterization of Bad(i, j) in terms of 'well-approximable' vectors in the area of 'twisted' inhomogeneous Diophantine approximation is established. In addition, it is shown that Bad x (i, j), the 'twisted' inhomogeneous analogue of Bad(i, j), has full Hausdorff dimension 2 when x is chosen from Bad(i, j). The main results naturally generalise the i = j = 1/2 work of Kurzweil.

There is no analogue to Jarník’s relation for twisted Diophantine approximation

“…It is thus natural to consider the following twisted exponents. See the paper of Harrap [3] for further twisted Diophantine approximation.…”

Diophantine approximation for products of linear maps — logarithmic improvements

“…Also with this choice of ρ(T ) = κ log log T , provided that κ is chosen sufficiently small, we obtain that log   T We note that the lattices Λ(u, v, 0, 0) with (u, v) ∈ R 2 are precisely the lattices in the orbit U Z 3 , where U is the expanding horospherical subgroup defined in (10). Hence, it follows from Proposition 7 that for almost every (u, v) ∈ R 2 , the lattice Λ(u, v, 0, 0) satisfies the assumption of Proposition 12.…”

“…An old argument of Cassels readily implies that for almost all (u, v, α, β) ∈ R 4 , lim inf q→∞ (log q) 2 q qu − α qv − β = 0 1 (see, for instance, [10,Theorem 3.3]). The case with α = 0 was investigated by Haynes, Jensen and Kristensen in [11].…”

Fully Inhomogeneous Multiplicative Diophantine Approximation of Badly Approximable Numbers

“…These types of results are referred to as twisted Diophantine approximation statements, see [1,12]. In fact, Tseng proved a fortiori that S β is winning in the sense of Schmidt [27].…”