Random Multiplicative Functions and Making Squares from Polynomial Values

Technical Summary: Random Multiplicative Functions and Making Squares from Polynomial Values

Problem Statement
This paper investigates the limiting distribution of partial sums of random multiplicative functions (RMFs) evaluated at polynomial values. Specifically, for a fixed polynomial $P(X) \in \mathbb{Z}[X]$ with a positive leading coefficient, the authors study the asymptotic behavior of the sum:
$$\sum_{n \le N} f(P(n))$$
where $f$ is a random multiplicative function. The study focuses on two specific types of RMFs:

1. Rademacher RMF: $f(p)$ are i.i.d. random variables uniformly distributed on $\{1, -1\}$, and $f(n)$ is defined multiplicatively with $f(n)=0$ if $n$ is not square-free (modeling the Möbius function).
2. Extended Rademacher RMF: Defined similarly, but $f(n)$ is defined completely multiplicatively (non-zero for all $n$).

The central question is whether these sums, when normalized, converge to a standard Gaussian distribution $\mathcal{N}(0, 1)$. While the Steinhaus case (complex unit circle values) was previously solved for any polynomial by Klurman–Shkredov–Xu, the Rademacher and extended Rademacher cases present significantly harder Diophantine challenges.

Methodology
The authors employ a probabilistic framework based on Martingale theory, specifically the McLeish Central Limit Theorem, to reduce the probabilistic question to a set of Diophantine counting problems. The core of the methodology involves establishing "paucity" phenomena: proving that the number of solutions to specific Diophantine equations is dominated by "diagonal" solutions (where the polynomial values are trivially related), while "off-diagonal" solutions are negligible.

Key technical tools include:

• Diophantine Geometry: Utilizing results from Hooley, Evertse–Silverman, and Reuss to bound the number of integral points on curves and surfaces defined by polynomial equations.
• Sieve Methods: Applying sieve bounds (e.g., Huxley, Bombieri–Pila) to control the distribution of square divisors in polynomial values.
• Pell–Fermat Equations: For the quadratic case ($\deg P = 2$), the authors develop a new method leveraging the multiplicative structure of quadratic norm forms and the size of solutions to Pell–Fermat equations to obtain sharper estimates.
• Combinatorial Decomposition: For higher degrees, the authors use a combinatorial decomposition of the solution set based on the complete graph $K_4$, labeling edges with greatest common divisors of polynomial values to manage large square factors.
• Numerical Optimization: The exponent in the fourth moment estimates for quadratic polynomials is optimized using linear programming.

Key Contributions and Results

1. Central Limit Theorems (CLT):

   • Theorem 1.1 (Rademacher Case): Proves a CLT for $\sum f(P(n))$ where $f$ is a Rademacher RMF, provided $P$ is separable, $\deg P \ge 2$, and the set of $n$ where $P(n)$ is square-free has positive lower density. This extends previous work by Chinis–Shala (which covered $\deg P = 2$) to general polynomials.
   • Theorem 1.2 (Extended Rademacher Case): Proves an unconditional CLT for $\sum g(P(n))$ where $g$ is an extended Rademacher RMF, assuming $P$ is separable, $\deg P \ge 2$, and every irreducible factor of $P$ has degree $\le 3$.

2. Diophantine Estimates:

   • Quadratic Case ($\deg P = 2$): The authors establish sharp bounds for the number of off-diagonal solutions to $P(n_1)P(n_2) = \square$ and $P(n_1)P(n_2)P(n_3)P(n_4) = \square$.
     • For the variance (second moment), the number of off-diagonal solutions is $O(\sqrt{N})$ (Proposition 5.1).
     • For the fourth moment, the number of off-diagonal solutions is $O(N^{3/2+\varepsilon})$ (Proposition 6.1). These bounds are crucial for verifying the CLT criteria.
   • Higher Degree Case ($\deg P \ge 3$): The authors provide estimates for the fourth moment under the assumption that irreducible factors have degree $\le 3$. They prove that the number of solutions to $P(n_1)P(n_2)P(n_3)P(n_4) = \square$ with bounded square-free cores is $O(A^{11/9}N^{14/9+\varepsilon})$ (Proposition 8.1).

3. Variance Computation:
   The paper highlights that computing the variance for the extended Rademacher case is non-trivial because the orthogonality present in the Steinhaus case does not hold. The variance depends on counting solutions to $P(n_1)P(n_2) = \square$, which requires distinguishing between square-free and non-square-free values of $P(n)$.

Significance and Claims
The authors state that their work solves the Diophantine problems required to establish CLTs for a large family of polynomials in both the Rademacher and extended Rademacher settings.

• Sharpness: The results are sharpest for $\deg P = 2$, where the theory of Pell–Fermat equations allows for tighter bounds than for $\deg P \ge 3$.
• Unconditional Results: Theorem 1.2 provides an unconditional Gaussian limit for polynomials with irreducible factors of degree $\le 3$.
• Limitations and Open Problems: The authors note that for polynomials with irreducible factors of degree $\ge 4$, their proofs currently rely on a power-saving version of the Square-Free Sieve Conjecture (specifically, the assumption $H_\delta$). They explicitly state that proving the CLT unconditionally for such polynomials remains an open problem.
• Diophantine Interest: Beyond the probabilistic applications, the paper presents the Diophantine equations $P(n_1)P(n_2) = \square$ and $P(n_1)P(n_2)P(n_3)P(n_4) = \square$ as interesting problems in their own right, offering new combinatorial decompositions and bounds that may inspire future advances in Diophantine analysis.

The paper acknowledges that while the probabilistic reduction to Diophantine questions is standard (via Martingale theory), the novelty lies in the specific Diophantine estimates required for the Rademacher and extended Rademacher cases, which are significantly more difficult than the Steinhaus case due to the lack of perfect orthogonality and the presence of square factors.