Let $\theta\in\mathbb Q\setminus\{0\}$. We construct an approximating function $\psi:\mathbb N\to[0,\frac12)$ for which \[ \sum_{q=1}^{\infty}\frac{\varphi(q)}{q}\psi(q)=\infty, \] but the set of $x\in[0,1]$ for which \[ |qx-a-\theta|<\psi(q),\qquad \gcd(a,q)=1, \] holds for infinitely many $(a,q)\in\mathbb Z\times\mat...
We establish the Hausdorff dimension for exactly approximable matrices for all dimensions, including the inhomogeneous case. More precisely, let $\psi:\N\to(0,\infty)$ be a non-increasing approximation function and let $\tau_\psi$ denote its lower order. For $m,n\ge1$ and $\boldsymbol\theta\in\mathbb R^m$, denote $\Exa...
Let $d\geq 1$. Let $K\subset\mathbb{R}^d$ be a non-singleton self-similar set generated by a finite strongly irreducible iterated function system satisfying the open set condition, and let $\delta=\dim_{\mathrm H} K$. For $\tau>1/d$, set \[ W_d(\tau) = \left\{ \mathbf{x}\in\mathbb{R}^d: |q\mathbf{x}-\mathbf{p}|<q^{-\ta...
Yu-Bin He, Ling-Min Liao· 0 citations
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