[Paper Review] Ubiquitous systems and metric number theory

Arnaud Durand|ArXiv.org|Sep 22, 2007

Mathematical Dynamics and Fractals17 references3 citations

TL;DR

This paper establishes that sets defined by inhomogeneous Diophantine approximation conditions—specifically, points in $$\mathbb{R}^d$$ approximated by a family of balls with radii raised to a power $t \geq 1$—belong to the class $\mathrm{G}^{h}(\mathbb{R}^d)$ of sets with large intersection properties with respect to a gauge function $h$, provided the underlying family forms a heterogeneous ubiquitous system. The key result is that such sets have infinite Hausdorff $g$-measure for all gauge functions $g$ increasing faster than $h$ near zero, extending classical results in metric number theory.

ABSTRACT

We investigate the size and large intersection properties of $$E_{t}=\{x\in\R^d \:|\: \|x-k-x_{i}\|0$. We establish that the set $E_{t}$ belongs to the class $\grint^h(\R^d)$ of sets with large intersection with respect to a certain gauge function $h$, provided that $(x_{i},r_{i})_{i\in I}$ is a heterogeneous ubiquitous system with respect to $μ$. In particular, $E_{t}$ has infinite Hausdorff $g$-measure for every gauge function $g$ that increases faster than $h$ in a neighborhood of zero. We also give several applications to metric number theory.

Motivation & Objective

• To characterize the size and large intersection properties of sets defined by inhomogeneous Diophantine approximation in $\mathbb{R}^d$.
• To extend classical results on Hausdorff dimension and measure to more general approximation settings involving heterogeneous ubiquitous systems.
• To prove that the set $E_t$ of $t$-approximable points belongs to the class $\mathrm{G}^{h}(\mathbb{R}^d)$ for an appropriate gauge function $h$, implying infinite $g$-measure for faster-growing $g$.
• To apply the framework to specific number-theoretic settings, including rationals with prime numerators/denominators and numbers with restricted digit frequencies.

Proposed method

• The paper uses the concept of a heterogeneous ubiquitous system with respect to a Borel measure $\mu$ and exponent $\alpha$, where the $\mu$-mass of balls centered at $x_i$ with radius $r_i$ behaves as $r_i^\alpha$.
• It introduces the set $E_t = \{x \in \mathbb{R}^d \mid \|x - k - x_i\| < r_i^t \text{ for infinitely many } (i,k)\}$, generalizing classical approximation sets.
• The main technical tool is the construction of a suitable gauge function $h_{\beta, \varphi}$, derived from the dimension and growth conditions of the system, to define the class $\mathrm{G}^h(\mathbb{R}^d)$.
• Lemmas are used to control the measure of balls under the given system, relying on the asymptotic frequency of digits or partial quotients in expansions.
• The proof leverages the invariance of the large intersection class under similarities and countable intersections, using a covering argument based on dyadic cubes and frequency conditions.
• The framework is applied to three cases: rationals with prime numerators/denominators, numbers with digit frequencies constrained by a probability vector $\pi$, and numbers with partial quotients satisfying a Birkhoff-type condition.

Experimental results

Research questions

• RQ1 Under what conditions does the set $E_t$ of $t$-approximable points in $\mathbb{R}^d$ have infinite Hausdorff $g$-measure for all $g \prec h$?
• RQ2 When does a family of balls indexed by $i \in I^{\mu,\alpha}$ form a heterogeneous ubiquitous system with respect to a measure $\mu$?
• RQ3 How do large intersection properties of $E_t$ depend on the growth rate of the radii $r_i$ and the distribution of centers $x_i$?
• RQ4 Can the large intersection class $\mathrm{G}^h(\mathbb{R}^d)$ be used to refine classical results on the size of approximation sets in metric number theory?
• RQ5 What is the role of digit frequency or partial quotient constraints in determining the gauge function $h$ for the large intersection class?

Key findings

• The set $E_t$ belongs to the class $\mathrm{G}^{h}(\mathbb{R}^d)$ for a gauge function $h = h_{\beta/t, 2\varphi + \phi/(1+d)}$ when $t > 1$, and $h = h_{\beta, \varphi}$ when $t = 1$, under the assumption of a heterogeneous ubiquitous system.
• For any gauge function $g$ such that $g \prec h$, the Hausdorff $g$-measure of $E_t$ is infinite, indicating maximal size in the sense of metric theory.
• The result applies to the set $J^\mathbb{P}_\tau$ of $\tau$-approximable numbers by rationals with prime numerators and denominators, showing it belongs to $\mathrm{G}^h(\mathbb{R})$ for $h$ satisfying $\sum_q h(q^{-\tau}) q / (\log q)^2 = \infty$.
• For numbers with digit frequencies constrained by a probability vector $\pi$, the set $E^\mathrm{Dev}_t$ belongs to $\mathrm{G}^{h_{\beta,\varphi}}((0,1)^d)$, with $h_{\beta,\varphi}$ depending on the entropy and frequency deviation.
• For numbers with partial quotients satisfying a Birkhoff-type condition, the set $E^\mathrm{Bir}_t$ belongs to $\mathrm{G}^{h_{\beta/t, 2\varphi + \phi}}((0,1)^d)$, with $\phi$ related to the growth of the Birkhoff sum.
• The framework unifies and generalizes previous results on Hausdorff measure and dimension for approximation sets, including Jarník’s theorem and Falconer’s large intersection property.