Uniform Diophantine approximation: improving Dirichlet’s theorem

Abstract: In this talk, I will discuss the metrical theory associated with the set of Dirichlet non-improvable numbers.

Let $\Psi :[1,\infty )\rightarrow \mathbb{R}_{+}$ 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 $\Psi $-Dirichlet non-improvable numbers $$ G(\Psi):=\Big\{x\in \lbrack 0,1):a_{n}(x)a_{n+1}(x)\,>\,\Psi \big(q_{n}(x) \big)\ \mathrm{for\ infinitely\ many}\ n\in \mathbb{N}\Big\}, $$ is related with the classical set of $1/q^{2}\Psi (q)$-approximable numbers

$$ \mathcal{K}(\Psi):=\left\{x\in[0,1): \left|x-\frac pq\right|<\frac{1}{ q^2\Psi(q)} \ \mathrm{for \ infinitely \ many \ } (p, q)\in \mathbb{Z}\times \mathbb{N } \right\}, $$ in the sense that $\mathcal{K}(3\Psi )\subset G(\Psi )$. In this talk, I will explain that the Hausdorff measure of the set $G(\Psi)$ obeys a zero-infinity law for a large class of dimension functions. Together with the Lebesgue measure-theoretic results established by Kleinbock \& Wadleigh (2016), our results contribute to building a complete metric theory for the set of Dirichlet non-improvable numbers.

Another recent result that I will discuss will be the Hausdorff dimension of the set $G(\Psi)\setminus \mathcal{K}(3\Psi )$.

number theory

Audience: researchers in the topic

Number Theory Online Conference 2020