Measure theoretic properties of large products of consecutive partial quotients

Technical Summary of "Measure Theoretic Properties of Large Products of Consecutive Partial Quotients"

Problem Statement
The paper investigates the measure-theoretic and fractal properties of sets of irrational numbers defined by the behavior of products of consecutive partial quotients in their regular continued fraction expansions. Specifically, for a non-decreasing function $\phi: \mathbb{N} \to \mathbb{R}_{\ge 2}$ and a fixed integer $\ell \ge 1$, the authors study the set $F_\ell(\phi)$ consisting of irrational numbers $x \in [0,1)$ such that, for infinitely many $n$, there exist two distinct indices $1 \le j < k \le n$ where the products of $\ell$ consecutive partial quotients starting at $j$ and $k$ both exceed $\phi(n)$:
$$a_j(x)\cdots a_{j+\ell-1}(x) \ge \phi(n) \quad \text{and} \quad a_k(x)\cdots a_{k+\ell-1}(x) \ge \phi(n).$$
This problem extends previous work on single partial quotients ($E_1(\phi)$) and products of two consecutive quotients ($E_2(\phi)$), as well as recent results concerning the existence of two large blocks ($F_1(\phi)$ and $F_2(\phi)$). The core challenge addressed is determining the Lebesgue measure and Hausdorff dimension of $F_\ell(\phi)$ for arbitrary $\ell \ge 2$, particularly handling the complex interactions arising from overlapping blocks of partial quotients.

Methodology
The authors employ a combination of ergodic theory, continued fraction dynamics, and analytic number theory.

1. Lebesgue Measure Analysis:

   • Convergence Case: To prove $m(F_\ell(\phi)) = 0$ when a specific series converges, the authors utilize the Borel-Cantelli Lemma. They estimate the measure of the sets where the condition holds at step $n$ by decomposing the problem into cases based on the distance between the two blocks of length $\ell$.
   • Divergence Case: To prove $m(F_\ell(\phi)) = 1$ when the series diverges, they employ the Chung-Erdős inequality to establish a lower bound for the measure of the limsup set. A crucial step involves showing that the set intersects a large portion of every fundamental interval, allowing the application of Knopp's Lemma.
   • Key Technical Tool: The proof relies heavily on new estimates for series involving products of partial quotients in overlapping blocks. Theorem 3.4 provides asymptotic bounds for sums of the form $\sum \frac{1}{a_1^2 \cdots a_r^2 b_1^2 \cdots b_j^2 c_1^2 \cdots c_r^2}$ subject to product constraints, which accounts for the combinatorial complexity of overlaps of lengths $0$ to $\ell-1$.

2. Hausdorff Dimension Analysis:

   • The authors utilize the thermodynamic formalism of the Gauss map $T$. The dimension is characterized via the pressure function $P(T, \psi)$, where the potential $\psi$ depends on the growth rate of $\phi$ (denoted by $B_\phi$) and the dimension parameter $s$.
   • Lower Bound: Constructed by defining a Cantor-like subset $S_\ell$ within $F_\ell(\phi)$ with specific constraints on the partial quotients. The Mass Distribution Principle is applied, and the dimension is maximized over a family of parameters $(x,y)$ representing the distribution of growth rates among the blocks.
   • Upper Bound: Established by constructing efficient covers of the set $F_\ell(\phi)$. The set is partitioned into subsets based on the length of the overlap between the two large product blocks (e.g., no overlap, overlap of length 1, etc.). For each subset, the authors derive upper bounds on the $s$-dimensional Hausdorff measure by summing the diameters of covering intervals raised to the power $s$, utilizing estimates from Lemma 3.9 and Lemma 3.10.

Key Contributions and Results

1. Generalization of Lebesgue Measure Results (Theorem 1.5):
   The paper determines the Lebesgue measure of $F_\ell(\phi)$ for any $\ell \ge 2$. The measure is 0 or 1 depending on the convergence or divergence of the series:
   $$\sum_{n=1}^\infty \left( \frac{n \log^{2(\ell-1)} \phi(n)}{\phi^2(n)} + \frac{\log^{\ell-2} \phi(n)}{\phi(n)} \right).$$
   The first term corresponds to non-overlapping blocks, while the second term accounts for overlapping blocks. This generalizes Theorem 1.3 ($\ell=1$) and Theorem 1.4 ($\ell=2$).

2. Computation of Hausdorff Dimension for $\ell=3$ (Theorem 1.10):
   The authors explicitly compute the Hausdorff dimension of $F_3(\phi)$. For $1 < B_\phi < \infty$, the dimension is given by:
   $$\dim_H F_3(\phi) = \inf \{ s \ge 0 : P(T, -g(s)\log B_\phi - s \log |T'|) \le 0 \},$$
   where $g(s) = \frac{3s^3 - 5s^2 + 4s - 1}{s^2 - s + 1}$. This result extends the known formulas for $\ell=1$ and $\ell=2$.

3. Implications for Strong Laws of Large Numbers:
   A consequence of the measure result is that the strong law of large numbers for products of $\ell$ consecutive partial quotients fails even if the block with the largest product is removed. Specifically, for almost every $x$, the limit superior of the normalized sum (minus the maximum term) is infinite.

Significance and Claims
The paper claims that Theorem 1.5 is a "crucial step" toward extending the laws of large numbers for sums of products of partial quotients to arbitrary $\ell$. The authors conjecture that, with these measure-theoretic tools, one can prove that for almost all $x$:
$$\lim_{n \to \infty} \frac{1}{n \log^\ell n} \left( \sum_{j=1}^n a_j(x)\cdots a_{j+\ell-1}(x) - \max_{1 \le k \le n} a_k(x)\cdots a_{k+\ell-1}(x) \right) = \frac{1}{\ell \log 2}.$$
The authors also note that the proof techniques for Theorem 1.10 appear adaptable to compute $\dim_H F_\ell(\phi)$ for $\ell \ge 4$, though they do not provide the explicit recursive formula for the potential in this paper. The work is presented as a foundational step in understanding the statistical behavior of large products in continued fractions, distinguishing between null sets via Hausdorff dimension and establishing the impossibility of certain strong laws of large numbers.