On Multiplicatively Badly Approximable Vectors
Reynold Fregoli, Dmitry Kleinbock
TL;DR
This paper extends multiplicative Diophantine approximation to higher dimensions by proving that the set of vectors in any box B⊂[0,1]^d that satisfy a positive lower bound for q(\log q)^{d-1}\log\log q times the product of fractional parts has full Hausdorff dimension. The authors introduce a new multiplicative Dani correspondence, along with ψ-approximability sets W_{m,n}^{×}(ψ) defined via a ψ-dependent cone C_R, linking Diophantine properties to dynamics on the space of unimodular lattices. The core strategy combines an induction on the ambient dimension d with a Cantor-type construction and a sophisticated counting-on-average argument for dangerous intervals, leveraging lattice-minima estimates and duality to control both dual and simultaneous cases. The result generalizes Badziahin’s 2D phenomenon to all dimensions d≥2 and yields a robust toolkit (Cantor-rich constructs, multiplicative Dani correspondence) that may prove useful for a broader class of multiplicative Diophantine problems. Overall, the paper provides a new proof technique and a higher-dimensional perspective on multiplicatively badly approximable vectors, with potential implications for related problems in dynamics and number theory.
Abstract
Let $\langle x\rangle$ denote the distance from $x\in\mathbb{R}$ to the set of integers $\mathbb{Z}$. The Littlewood Conjecture states that for all pairs $(α,β)\in\mathbb{R}^{2}$ the product $q\langle qα\rangle\langle qβ\rangle$ attains values arbitrarily close to $0$ as $q\in\mathbb{N}$ tends to infinity. Badziahin showed that if a factor $\log q\cdot \log\log q$ is added to the product, the same statement becomes false. In this paper, we generalise Badziahin's result to vectors $\boldsymbolα\in\mathbb{R}^{d}$, replacing the function $\log q\cdot \log\log q$ by $(\log q)^{d-1}\cdot\log\log q$ for any $d\geq 2$, and thereby obtaining a new proof in the case $d=2$. Our approach is based on a new version of the well-known Dani Correspondence between Diophantine approximation and dynamics on the space of lattices, especially adapted to the study of products of rational approximations. We believe that this correspondence is of independent interest.