Multiplicative functions additive on partitions of $2k$ nonzero squares

Technical Summary: Multiplicative Functions Additive on Partitions of $2k$ Nonzero Squares

Problem Statement
This paper investigates the classification of multiplicative functions $f: \mathbb{N} \to \mathbb{C}$ that satisfy a specific additivity condition on partitions of integers into $2k$ nonzero squares. Specifically, for a fixed integer $k \ge 3$, the study focuses on functions satisfying:
$$f\left( \sum_{i=1}^{2k} x_i^2 \right) = \sum_{j=1}^{k} f(x_{2j-1}^2 + x_{2j}^2)$$
for all positive integers $x_1, \dots, x_{2k}$.

This problem is framed as a higher-order analogue to a result by Park [10], which addressed the case $k=2$ (sums of two pairs of squares). The condition is equivalent to requiring that $f$ is $k$-additive on the set $E_2$, where $E_2 = \{a^2 + b^2 : a, b \ge 1\}$ is the set of sums of two nonzero squares. The central question is whether such functions are forced to be the identity function ($f(n)=n$) or if non-trivial exceptional solutions exist.

Methodology
The analysis relies on the interplay between the multiplicative property of $f$ and the additive structure of integers representable as sums of $2k$ nonzero squares. The methodology proceeds in three main stages:

1. Representability Results: The proofs utilize Dubouis's theorem [4] regarding the representation of integers as sums of $m$ nonzero squares. A critical distinction is made between small $k$ and large $k$:

   • For $k=2$, the set of sums of $2k$ squares excludes infinite arithmetic progressions (e.g., $\{2 \cdot 4^r, 6 \cdot 4^r, \dots\}$), complicating the analysis.
   • For $k \ge 3$, the set of sums of $2k$ nonzero squares contains all integers greater than a specific threshold (specifically, all $n \ge 20$ for $k=3$ and $n \ge 22$ for $k=4$). This "almost all" property allows for strong induction arguments.

2. Base Case Evaluation: For specific values of $k$, the authors evaluate $f$ on small integers by constructing specific decompositions of numbers into elements of $E_2$. By equating different decompositions of the same integer (e.g., $18 = 2+8+8$ vs. $18 = 2+2+14$), they derive systems of equations involving $f(2), f(3), f(5)$, etc. The non-degeneracy condition $f(2) \neq 0$ is essential to rule out the trivial zero function (where $f(n)=0$ for $n>1$).

3. Inductive and Asymptotic Arguments:

   • For $k=3, 4$: The authors use strong induction. Once $f(n)=n$ is established for an initial segment of integers, any integer $N$ beyond this segment can be written as a sum of $k$ elements from $E_2$, all strictly smaller than $N$. The $k$-additivity condition then forces $f(N)=N$.
   • For $k \ge 5$: The approach shifts to analyzing the asymptotic behavior. By comparing two specific decompositions of consecutive integers $N$ and $N+1$ (using the fact that $12=2+10$ and $13=5+8$ are in $E_2+E_2$), the authors show that the difference $f(N+1) - f(N)$ becomes constant for sufficiently large $N$. This implies $f(n)$ is eventually linear ($f(n) = cn+d$). Multiplicativity constraints on large primes then force the coefficients $(c,d)$ to be either $(1,0)$ (identity) or $(0,0)$ (zero function).

Key Contributions and Results

• Theorem 1.1 ($k=3, 4$): If $f$ is a multiplicative function satisfying the condition for $k \in \{3, 4\}$ and $f(2) \neq 0$, then $f(n) = n$ for all positive integers $n$. The paper proves that the only obstruction to uniqueness is the vanishing of $f(2)$, which allows for the trivial zero solution.
• Theorem 1.2 ($k \ge 5$): If $f$ is multiplicative and satisfies the condition for $k \ge 5$, then $f$ must be either the identity function on $\mathbb{N}$, or $f(n) = 0$ for all $n > 2k + 21$.
• Corollary 5.1: For $k \ge 5$, if there exists any $a > 2k + 21$ such that $f(a) \neq 0$, then $f$ is the identity function on all of $\mathbb{N}$.

Significance and Claims
The paper claims to extend the scope of additive uniqueness sets for multiplicative functions. While previous work (Park [10]) established that sums of two nonzero squares form a 2-additive uniqueness set (with specific exceptions), this work demonstrates that for higher $k$, the set of sums of $2k$ nonzero squares acts as a $k$-additive uniqueness set under mild non-degeneracy conditions.

The authors highlight that the transition from $k=2$ to $k \ge 3$ is driven by a structural change in the representability of integers: the set of sums of $2k$ nonzero squares becomes co-finite (containing all but finitely many integers) for $k \ge 3$. This property simplifies the classification, allowing the authors to prove that the identity function is the unique non-trivial solution for $k=3,4$, and that solutions must eventually degenerate to zero or the identity for $k \ge 5$.

The paper notes that the threshold $k=5$ in Theorem 1.2 is sharp for the specific method employed, as the proof relies on the sumset of $2(k-2)$ nonzero squares containing all but finitely many integers, a property that fails for $k=4$ due to the infinite exceptional progressions identified in Dubouis's theorem.