5 problems

Let $d=2$. A weight vector is a vector $\mathbf{w}=(w_1,w_2)\in\mathbb{R}^2$ with $w_i\geq 0$ and $w_1+w_2=1$. For a weight vector $\mathbf{w}$, let … where $|\cdot|_{\mathbb{Z}}$…

Countability conjecture. If $x\in\mathbb{Q}^d$, then $\Phi(x)$ is countable.

Let $\mathbf{w}=(w_1,\ldots,w_d)\in\mathbb{R}^d$ be a weight vector, meaning that $w_i\geq 0$ for $1\leq i\leq d$ and $\sum_{i=1}^d w_i=1$. Define … where $|\cdot|_{\mathbb{Z}}$ de…

Most badly approximable 2-tuple conjecture. The “most” badly approximable 2-tuple is

Let real two-dimensional weighted badly approximable vectors be defined with respect to two weightings of the approximation problem. Schmidt's intersection conjecture. Any two sets…