Intrinsic Diophantine approximation: a solution to Mahler's problem

1. Problem Statement and Context

The paper addresses the problem of intrinsic Diophantine approximation on self-similar fractals, specifically focusing on the middle-third Cantor set ($K_{1/3}$) and general homogeneous rational Iterated Function Systems (IFS).

• The Core Question: How well can points $x$ within a fractal set $K$ be approximated by rational numbers $r$ that also lie inside $K$? This is distinct from "extrinsic" approximation, where rationals can be anywhere in $\mathbb{R}$.
• The Specific Constraint: The paper investigates approximation by rationals $r = p/q$ where the denominator $q$ (or the height $h(r)$) has a bounded number of distinct prime factors. Let $PN = \{q \in \mathbb{N} : \omega(q) \le N\}$, where $\omega(q)$ is the number of distinct prime factors of $q$.
• The Conjecture: The authors aim to verify a conjecture extending the work of Bugeaud and Durand. The conjecture posits that for a non-increasing approximation function $\psi$, the Hausdorff dimension of the set of points approximable at rate $\psi$ is:
  $$\dim_H \{x \in K : |x - r| \le \psi(h(r)) \text{ i.o. } r \in \mathbb{Q} \cap K\} = \frac{\dim_H K}{\max\{1, \delta_\psi\}}$$
  where $\delta_\psi = \liminf_{q \to \infty} \frac{-\log \psi(q)}{\log q}$ is the shrinking rate.

2. Methodology

The paper employs a combination of geometric measure theory, ergodic theory, and analytic number theory.

A. Geometric Framework

• IFS and Attractors: The study focuses on IFSs $T = \{f_i(x) = \frac{x+p_i}{b}\}_{i=1}^m$ with rational parameters. The attractor $K_T$ is a self-similar set.
• Separation Conditions: The authors utilize the Weak Separation Condition (WSC) and the Asymptotically Weak Separation Condition (AWSC). They prove that rational IFSs satisfy these conditions, which implies the attractor is Ahlfors regular (a crucial property for dimension calculations).
• Intrinsic Height: For inhomogeneous IFSs, the paper uses the "intrinsic height" $h_{int}(r)$, defined based on the periodic coding of the rational $r$ within the IFS, rather than the standard denominator.

B. Upper Bound Strategy (The "Hard" Part)

To prove the upper bound for the dimension, the authors must show that there are "not enough" rational points in the fractal to cover the set efficiently.

• Reduction to Torus Dynamics: The problem is projected onto the 1-torus $\mathbb{T}^1$. The set of rationals in the fractal corresponds to a $\times b$-invariant set $S$.
• Equidistribution and Multiplicative Order: The core difficulty is that if a rational $p/q$ is in the fractal, the sequence $b^k p \pmod q$ must stay within the fractal's projection. If the fractal has empty interior, this sequence cannot be equidistributed modulo $q$ unless the multiplicative order of $b$ modulo $q$ (denoted $O_q(b)$) is very small.
• Counting Rationals: The authors use Bourgain's Theorem on exponential sums to show that if $O_q(b)$ is large, the sequence equidistributes, forcing the rational out of the fractal (since the fractal's complement contains an interval).
• Number Theoretic Estimates: They establish that the number of denominators $q \le x$ with a small multiplicative order and a bounded number of prime factors is negligible (logarithmically small). This leads to the conclusion that the "density" of such rationals is insufficient to increase the dimension beyond the intrinsic limit.

C. Lower Bound Strategy

• Mass Transference Principle: The lower bound is derived using a mass transference principle (from [12]), which transfers measure-theoretic results (full measure) to dimension results.
• Periodic Sequences: The authors construct a limsup set based on eventually periodic sequences in the IFS coding. They show that the set of points approximable by these periodic points (which correspond to rationals) has the desired dimension.

3. Key Contributions and Results

Theorem 1.1 (Main Result for Homogeneous Rational IFS)

For a homogeneous rational IFS $T$ with attractor $K_T$ (where $K_T$ has empty interior), and for any $N \ge \omega(b) + \omega(b-1)$:
$$\dim_H \{x \in K_T : |x - r| \le \psi(h(r)) \text{ i.o. } r \in \mathbb{Q} \cap K_T, h(r) \in P_N\} = \frac{\dim_H K_T}{\max\{1, \delta_\psi\}}$$

• Significance: This confirms the conjecture for rationals with a bounded number of prime factors. It shows that restricting the denominators to $P_N$ does not change the dimension formula, provided $N$ is large enough.
• Special Case: For the middle-third Cantor set ($b=3$), since $\omega(3)=1$ and $\omega(2)=1$, the result holds for any $N \ge 1$.

Theorem 1.5 (Conditional Result for Full Set of Rationals)

If Conjecture 1.4 holds (a number-theoretic conjecture regarding the distribution of multiplicative orders), then the formula holds for the entire set of rationals in $K_T$ (without the $P_N$ restriction):
$$\dim_H \{x \in K_T : |x - r| \le \psi(h(r)) \text{ i.o. } r \in \mathbb{Q} \cap K_T\} = \frac{\dim_H K_T}{\max\{1, \delta_\psi\}}$$

• Conjecture 1.4: States that the set of $q$ where the multiplicative order $O_q(b)$ is unusually small ($O_q(b) \le q^{f(q)}$) has logarithmic density zero.

Theorem 1.8 (Generalization to Inhomogeneous IFS)

For general rational IFSs (where contraction ratios $1/q_i$ may differ), the authors prove the dimension formula using the intrinsic height $h_{int}(r)$:
$$\dim_H \{x \in K_T : |x - r| \le \psi(h_{int}(r)) \text{ i.o. } r \in \mathbb{Q} \cap K_T\} = \frac{\dim_H K_T}{\max\{1, \delta_\psi\}}$$
This result relies on Theorem 4.1, which establishes the dimension for approximation by eventually periodic sequences on self-similar fractals satisfying the AWSC.

Corollary 1.9 (Extrinsic vs. Intrinsic)

Since $h_{int}(r) \ge h(r)$, the result provides a lower bound for approximation by standard height:
$$\dim_H \{x \in K_T : |x - p/q| \le \psi(q) \text{ i.o. } p/q \in \mathbb{Q} \cap K_T\} \ge \frac{\dim_H K_T}{\max\{1, \delta_\psi\}}$$

4. Significance and Implications

1. Resolution of a Major Conjecture: The paper makes significant progress toward the Bugeaud-Durand conjecture regarding intrinsic Diophantine approximation. It proves the formula for a dense subset of rationals (those with bounded prime factors) and reduces the general case to a specific number-theoretic conjecture.
2. Dichotomy of Interior vs. Empty Interior: The paper highlights a sharp dichotomy. If the fractal has non-empty interior, the dimension behaves like a standard Jarník set (dimension $\min\{1, s_\psi\}$). If the interior is empty, the dimension is governed by the intrinsic geometry of the fractal ($\dim_H K / \delta_\psi$).
3. Connection to Number Theory: The work bridges fractal geometry and analytic number theory. The proof relies heavily on the distribution of multiplicative orders modulo $q$ and the properties of primes, demonstrating that the geometric structure of fractals is deeply linked to the arithmetic properties of their defining parameters.
4. Generalization: By moving from homogeneous to inhomogeneous IFS and introducing the intrinsic height, the results apply to a much broader class of self-similar sets than previous works (which were often limited to the middle-third Cantor set or specific homogeneous cases).

5. Conclusion

E. Daviaud successfully establishes that for self-similar sets with empty interior defined by rational parameters, the Hausdorff dimension of the set of intrinsically approximable points is determined solely by the shrinking rate of the approximation function and the dimension of the set itself, provided the approximating rationals have a bounded number of distinct prime factors. The paper effectively isolates the arithmetic difficulty of the problem to the distribution of multiplicative orders, offering a clear path for resolving the full conjecture.