8 problems

Let $\mathbf{x}=(x_1,\ldots,x_k)\in\mathbb{R}^k$, where $k\geqslant2$. Littlewood's conjecture in $k$ dimensions. … This is the multiplicative form of simultaneous Diophantine appr…

Let $H(\mathbf m)$ denote the multiplicative height of $\mathbf m \in \mathbb Z^d$, and let ${\boldsymbol{\alpha}} \in \mathbb R^d$ with $d \geqslant 2$. Dual form of Littlewood's…

Beresnevich–Haynes–Velani conjecture. One has

Non-Liouville inhomogeneous multiplicative approximation conjecture. If

For $\lambda eu0007\mathbb{R}$, let $\operatorname{Mad}^{\lambda}(2,1)$ denote the corresponding logarithmic approximation set in the multiplicative setting. Badziahin–Velani conjectur…

Inhomogeneous multiplicative Khintchine-type conjecture. Then for almost all $\alpha_k\in\mathbb R$, there exist infinitely many $(n_1,\ldots,n_k)\in\mathbb Z^k$ such that

Higher-dimensional Hausdorff-measure divergence conjecture. If $\Psi(\mathbf q)=\psi(|\mathbf q|)$ for a monotonically decreasing $\psi:\mathbb N\to(0,\infty)$, this series diverge…

Let $m,d\geq1$ be integers, let $\Psi:\mathbb Z^m\to(0,\infty)$, and fix $\bftheta\in\I^d$. For $X=(\mathbf x^{(1)},\ldots,\mathbf x^{(d)})\in\I^{md}$, define…