Weighted twisted inhomogeneous diophantine approximation

Abstract

We prove a multidimensional weighted analogue of the well-known theorem of Kurzweil (1955) in the metric theory of inhomogeneous Diophantine approximation. Let ∑ i = 1 m α i = m and | ⋅ | α = max 1 ⩽ i ⩽ m | ⋅ | 1 / α i . Given an n-tuple of monotonically decreasing functions Ψ = ( ψ 1 , … , ψ n ) with ψ i : R + → R + such that each ψ i ( r ) → 0 as r → ∞ and fixed A ∈ R n × m define W A ( Ψ ) := { b ∈ [ 0 , 1 ] n : | A i ⋅ q − b i − p i | < ψ i ( | q | α ) ( 1 ⩽ i ⩽ n ) , for infinitely many ( p , q ) ∈ Z n × ( Z m ∖ { 0 } ) } . We prove that the set W A ( Ψ ) has zero-full Lebesgue measure under convergent–divergent sum conditions with some mild assumptions on A and the approximating functions Ψ. We also prove the Hausdorff dimension results for this set. Along with some geometric arguments, the main ingredients are the weighted ubiquity and weighted mass transference principle introduced recently by Kleinbock & Wang (2023 Adv. Math. 428 109154), and Wang & Wu (2021 Math. Ann. 381 243–317) respectively.