arXiv subjects

Philip Bos

Publications and source records attributed to Philip Bos.

3 recordsLinked to original sources

arXiv2022

The generalised Hausdorff measure of sets of Dirichlet non-improvable numbers

Let $ψ:\mathbb R_+\to\mathbb R_+$ be a non-increasing function. A real number $x$ is said to be $ψ$-Dirichlet improvable if the system $$|qx-p|< \, ψ(t) \ \ {\text{and}} \ \ |q|<t$$ has a non-trivial integer solution for all large enough $t$. Denote the collection of such points by $D(ψ)$. In this paper, we prove a zero-infinity law valid for all dimension functions under natural non-restrictive conditions. Some of the consequences are zero-infinity laws, for all essentially sub-linear dimension functions proved by Hussain-Kleinbock-Wadleigh-Wang (2018), for some non-essentially sub-linear dimension functions, and for all dimension functions but with a growth condition on the approximating function.

math.NT↗

arXiv2019

Hausdorff dimension of a set in the theory of continued fractions

In this article we calculate the Hausdorff dimension of the set \begin{equation*} \mathcal{F}(Φ)=\left\{ x\in \lbrack 0,1):\begin{aligned}a_{n+1}(x)a_n(x) \geq Φ(n) \ {\rm for \ infinitely \ many \ } n\in \mathbb N \ {\rm and } \\ a_{n+1}(x)< Φ(n) \ {\rm for \ all \ sufficiently \ large \ } n\in \mathbb N \end{aligned}\right\} \end{equation*} where $Φ:\mathbb{N}\rightarrow (1,\infty)$ is any function with $\lim_{n\to \infty} Φ(n)=\infty.$ This in turn contributes to the metrical theory of continued fractions as well as gives insights about the set of Dirichlet non-improvable numbers.

math.DS↗

arXiv2019

The sets of Dirichlet non-improvable numbers vs well-approximable numbers

Let $Ψ:[1,\infty )\rightarrow \mathbb{R}_{+}$ be a non-decreasing function, $a_{n}(x)$ the $n$'{th} partial quotient of $x$ and $q_{n}(x)$ the denominator of the $n$'{th} convergent. The set of $Ψ$-Dirichlet non-improvable numbers \begin{equation*} G(Ψ):=\Big\{x\in \lbrack 0,1):a_{n}(x)a_{n+1}(x)\,>\,Ψ\big(q_{n}(x) \big)\ \mathrm{for\ infinitely\ many}\ n\in \mathbb{N}\Big\}, \end{equation*} is related with the classical set of $1/q^{2}Ψ(q)$-approximable numbers $ \mathcal{K}(Ψ)$ in the sense that $\mathcal{K}(3Ψ)\subset G(Ψ)$. Both of these sets enjoy the same $s$-dimensional Hausdorff measure criterion for $s\in (0,1)$. We prove that the set $G(Ψ)\setminus \mathcal{K}(3Ψ)$ is uncountable by proving that its Hausdorff dimension is the same as that for the sets $\mathcal{K}(Ψ)$ and $G(Ψ)$. This gives an affirmative answer to a question raised by Hussain-Kleinbock-Wadleigh-Wang (2018).

math.NT↗