Badly approximable points in twisted Diophantine approximation and Hausdorff dimension

Abstract

For any j 1 , . . . , j n > 0 with n i=1 j i = 1 and any θ ∈ R n , let Bad θ (j 1 , . . . , j n ) denote the set of points η ∈ R n for which max 1≤i≤n ( qθ i − η i 1/ji ) > c/q for some positive constant c = c(η) and all q ∈ N. These sets are the 'twisted' inhomogeneous analogue of Bad(j 1 , . . . , j n ) in the theory of simultaneous Diophantine approximation. It has been shown that they have full Hausdorff dimension in the non-weighted setting, i.e provided that j i = 1/n, and in the weighted setting when θ is chosen from Bad(j 1 , . . . , j n ). We generalise these results proving the full Hausdorff dimension in the weighted setting without any condition on θ. Moreover, we prove dim(Bad θ (j 1 , . . . , j n ) ∩ Bad(1, 0, . . . , 0) ∩ . . . ∩ Bad(0, . . . , 0, 1)) = n.

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].…”

“…Tseng [30] established that if , then , where These types of results are referred to as twisted Diophantine approximation statements, see [1, 12]. In fact, Tseng proved a fortiori that is winning in the sense of Schmidt [27].…”

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.…”

Dimension estimates for badly approximable affine forms

“…On the other hand, the set Bad A also has full Hausdorff dimension for every A [BHKV10]. See [Har12,HM17,BM17] for the weighted setting.…”