14 problems

Beresnevich–Haynes–Velani conjecture. One has

For $lambda eu0007mathbb{R}$, let $operatorname{Mad}^{lambda}(2,1)$ denote the corresponding logarithmic approximation set in the multiplicative setting. Badziahin–Velani conjectur…

Hussain–Simmons divergence conjecture. If this series diverges, then

Hussain–Simmons conjecture. The Lebesgue measure of $\mathcal M^\times(n,m;\psi)$ depends on the convergence or divergence of this series.

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…

Let $b1,b2\that b5b4b1$?

Beresnevich–Velani conjecture. The minimal growth rate of $f$, when $d=2$, is conjectured to be of order $\log x$.

Non-Liouville inhomogeneous multiplicative approximation conjecture. If

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…

Let $m\geq2$ and let $\psi:\mathbb{Z}\backslash\{0\}\rightarrow[0,\frac{1}{2}]$. The set $\mathcal{C}_{1,m}^{\times}(\psi)$ is the multiplicative simultaneous-approximation set con…

For $\lambda\geq 0$, define … Here $\|x\|$ denotes the distance to the nearest integer, and $\dim$ denotes Hausdorff dimension. Badziahin–Velani conjecture. … and … This conjecture…

Let $p$ be a prime, let $m\in\mathbb{N}$, and let $\|\,\cdot\,\|$ denote distance to the nearest integer. Higher-dimensional mixed Littlewood conjecture. For $m$ sufficiently large…