Technical Summary: Positive Logarithmic Hausdorff Measures of Exceptional Sets for the $p$-adic and $t$-adic Littlewood Conjectures

1. Problem Statement

The paper addresses the metric properties of the exceptional sets associated with two major conjectures in Diophantine approximation: the $p$-adic Littlewood conjecture and the $t$-adic Littlewood conjecture.

• The $p$-adic Littlewood Conjecture: Introduced by de Mathan and Teulié, it posits that for every real number $x \in \mathbb{R}$ and prime $p$,
  $$\liminf_{q \to \infty} q |q|_p \|qx\| = 0,$$
  where $| \cdot |_p$ is the normalized $p$-adic absolute value and $\| \cdot \|$ is the distance to the nearest integer.
• The $t$-adic Littlewood Conjecture: A function field analogue over a finite field $\mathbb{F}_q$, asserting that for every formal Laurent series $\Theta \in \mathbb{F}_q((t^{-1}))$,
  $$\inf_{0 \neq Q \in \mathbb{F}_q[t]} |Q| |Q|_t |\langle Q\Theta \rangle| = 0,$$
  where $|Q| = q^{\deg Q}$, $|Q|_t = q^{-v_t(Q)}$, and $\langle \cdot \rangle$ denotes the fractional part.

It is known that if counterexamples exist, they must belong to the set of badly approximable numbers, which has Lebesgue measure zero. Furthermore, previous results by Einsiedler, Katok, Lindenstrauss, and Kleinbock established that the Hausdorff dimension of these exceptional sets is zero. The central question this paper addresses is: If these exceptional sets are non-empty, how "large" are they in a finer metric sense? Specifically, the authors investigate their size with respect to logarithmic Hausdorff measures, which are more sensitive than standard power-law measures when the dimension is zero.

2. Methodology

The authors employ a combination of dynamical systems theory, symbolic dynamics, and mass distribution principles to derive lower bounds on the Hausdorff measures of the exceptional sets.

2.1 Dynamical Formulation

The problem is reformulated in terms of the orbits of specific dynamical maps:

• For the $p$-adic case, the map is $T_p(y) = py \pmod 1$ on the circle $\mathbb{T} = \mathbb{R}/\mathbb{Z}$. The exceptional set $E_p$ is related to the orbit closures $F_p(x)$ of counterexamples.
• For the $t$-adic case, the map is the one-sided left shift $T$ on the space of Laurent series $I_q = t^{-1}\mathbb{F}_q[[t^{-1}]]$. The exceptional set $E_q^{(t)}$ is analyzed via the orbit closures $K(\Theta)$.

2.2 Uniform Diophantine Estimates

A core technical step involves establishing a uniform lower bound on the approximation quality for all points in the orbit closure of a counterexample.

• If $x$ is a counterexample with $m_p(x) > 0$, the authors prove that for all $y$ in the closure of the orbit of $x$, and for all integers $q$, the quantity $q|q|_p \|qy\|$ is bounded below by a positive constant $\beta$.
• Similarly, for the $t$-adic case, a uniform lower bound $c$ is established for the orbit of any counterexample $\Theta$.

2.3 Return Times and Cylinder Separation

The uniform Diophantine bounds are translated into constraints on the return times to cylinders in the symbolic representation (base-$p$ expansion or Laurent series coefficients).

• Lemma 2.7 (Return Times): If a point $y$ and its iterate $T^d y$ fall into the same cylinder of order $n$, the shift $d$ must satisfy $n < d + \text{const}$.
• This implies that the preimages of a cylinder under the map are pairwise disjoint for a sufficiently large number of iterations. This separation prevents the mass of the orbit from concentrating too heavily on small intervals.

2.4 Mass Distribution Principle

The authors construct invariant Borel probability measures supported on the orbit closures (using weak limits of empirical measures). By combining the disjointness of preimages with the uniform bounds, they derive an upper bound on the measure of any set $U$ of small diameter $r$:
$$\mu(U) \lesssim \frac{1}{\log(1/r)}.$$
Applying the Mass Distribution Principle (Lemma 2.1), this upper bound on the measure implies a lower bound on the Hausdorff measure with respect to the gauge function $h(r) = 1/\log(1/r)$.

2.5 Explicit Constructions (Odd Characteristic)

For the $t$-adic conjecture over fields with odd characteristic, the authors refine their results by utilizing the explicit counterexample construction by Lai and Sprang. This construction allows for a more precise analysis of the "branching" of the set at different scales, leading to a stronger gauge function with a higher exponent.

3. Key Contributions and Results

3.1 Results for the $p$-adic Littlewood Conjecture

Theorem 2.2: If the set of counterexamples $E_p$ is non-empty, then for any $x \in E_p$, the orbit closure $F_p(x)$ has positive logarithmic Hausdorff measure with respect to the gauge $h_1(r) = 1/\log(1/r)$.

• Corollary 2.3: Consequently, if $E_p$ is non-empty, it has the cardinality of the continuum.
• Corollary 2.4: The logarithmic Hausdorff dimension of $E_p$ is at least 1 ($\dim_{\log} E_p \ge 1$).

The authors note that this does not contradict the zero Hausdorff dimension results of Einsiedler et al., as a set can have zero dimension but positive measure with respect to a logarithmic gauge.

3.2 Results for the $t$-adic Littlewood Conjecture

Theorem 3.1: For any prime power $q$, if the exceptional set $E_q^{(t)}$ is non-empty, then:

1. The orbit closure of any counterexample has positive $h_1$-measure.
2. The entire exceptional set $E_q^{(t)}$ has infinite $h_1$-measure.
3. $E_q^{(t)}$ has cardinality continuum and logarithmic dimension at least 1.

Theorem 3.2 (Stronger Result for Odd $q$): Utilizing the Lai-Sprang construction for odd $q$, the authors prove a significantly stronger result. Let $A_q = \frac{q-1}{2} \log_2(q-1)$.

• There exists a subset $C_q \subset E_q^{(t)}$ such that $0 < H_{h_{A_q}}(C_q) < \infty$, where $h_{A_q}(r) = 1/(\log(1/r))^{A_q}$.
• The full exceptional set satisfies $H_{h_{A_q}}(E_q^{(t)}) = \infty$.
• This implies $\dim_{\log} E_q^{(t)} \ge A_q$.
• Note: For $q=3$, $A_3=1$. For $q \ge 5$, $A_q > 1$.

Characteristic 2: The paper notes that for characteristic 2, the conclusion with exponent 1 remains conditional on the existence of a counterexample, as the specific construction used for odd characteristics does not directly apply.

4. Significance and Claims

The paper claims to approach the Littlewood conjectures from a "metric" perspective, complementing the existing "dimension" results.

• Refining the "Smallness" of Counterexamples: While previous work showed that if counterexamples exist, they form a set of Hausdorff dimension zero, this paper demonstrates that such a set cannot be "too small." Specifically, it cannot be countable, and it must carry positive (or infinite) measure under logarithmic gauges.
• Cardinality: The results definitively establish that any non-empty exceptional set must have the cardinality of the continuum.
• Potential Path to Proof: The authors suggest a potential strategy for proving the conjectures themselves. If one could prove that the exceptional set has zero measure with respect to a specific gauge $h$ (via upper bound methods) while simultaneously proving that non-emptiness implies positive (or infinite) measure with respect to the same gauge (via lower bound methods), the set would be forced to be empty. The paper provides the necessary lower bound machinery for logarithmic gauges.
• Optimality: The paper modestly frames its results as lower bounds. It poses the question of whether these bounds can be strengthened (e.g., by increasing the exponent $A$ in the gauge function) or whether analogous results hold for the classical Littlewood conjecture.

In summary, the paper establishes that the exceptional sets for the $p$-adic and $t$-adic Littlewood conjectures, if they exist, are metrically substantial in the sense of logarithmic Hausdorff measures, possessing infinite cardinality and specific lower bounds on their logarithmic dimension.