The Minimal Absolute Value of Sums of Fifth Roots of Unity

Technical Summary: The Minimal Absolute Value of Sums of Fifth Roots of Unity

Problem Statement
This paper addresses the determination of the minimal absolute value of a non-vanishing sum of $n$ fifth roots of unity, chosen with repetition. Formally, for an integer $n \ge 1$, the authors define $\Sigma_5(n)$ as the set of sums $s = \sum_{i=0}^4 a_i \zeta^i$ where $a_i \in \mathbb{N}$ and $\sum a_i = n$, with $\zeta = \exp(2\pi i/5)$. The objective is to compute:
$$\sigma_5(n) = \min \{ |s| : s \in \Sigma_5(n) \setminus \{0\} \}$$
While the problem of determining $\sigma_\ell(n)$ for fixed $\ell$ and arbitrary $n$ is natural, it had not been previously considered in this specific form. Previous literature has focused on asymptotic formulas for fixed $n$ and large $\ell$, or on the characterization of weights $n$ for which vanishing sums exist (i.e., $0 \in \Sigma_\ell(n)$). The case $\ell=5$ is significant because the minimal polynomial of $\zeta_5$ is quartic, representing the smallest case where the degree of the minimal polynomial exceeds 2.

Methodology
The authors reduce the geometric problem of minimizing the absolute value of complex sums to a series of Diophantine approximation problems involving the golden ratio $\phi = (1+\sqrt{5})/2$ and the constant $\alpha = \phi + 2$.

1. Reduction to Real and Imaginary Parts: By expressing the sum $s$ in terms of its real and imaginary components, the authors relate $|s|$ to the distance of integer multiples of $\alpha$ and $\phi$ to the nearest integers. Specifically, they utilize the identities $\zeta + \zeta^4 = 1/\phi$ and $\zeta^2 + \zeta^3 = -\phi$.
2. Ostrowski Representations: The core analytical tool is the theory of continued fractions and Ostrowski representations. The authors utilize the fact that the continued fraction expansions of $\phi$ and $\alpha$ consist entirely of 1s (after the initial term for $\alpha$). This allows them to express integers uniquely as sums of non-consecutive Fibonacci numbers (for $\phi$) or Lucas numbers (for $\alpha$).
3. Bounding via Convergents: The proof strategy involves establishing lower bounds for $|s|$ by analyzing the "difference" terms $D_k = B_k t - A_k$ associated with the convergents of $\phi$ and $\alpha$. Using properties of these differences and the constraints imposed by the weight $n$, the authors derive strict inequalities that limit the possible values of the coefficients in the sum.
4. Case Analysis: The problem is split based on the congruence of $n$ modulo 5.
   • For $n \not\equiv 0 \pmod 5$, the analysis focuses on intervals defined by Lucas numbers ($L_m$) and twice Lucas numbers ($2L_m$).
   • For $n \equiv 0 \pmod 5$, the analysis focuses on intervals defined by Fibonacci numbers ($F_m$).

Key Contributions and Results

The paper provides exact formulas for $\sigma_5(n)$ for all $n \ge 1$, characterized by the Fibonacci ($F_m$) and Lucas ($L_m$) sequences.

1. Case $n \not\equiv 0 \pmod 5$ (Theorem 1)
For $n$ not divisible by 5, the minimal absolute value is given by:
$$\sigma_5(n) = \frac{1}{\phi^{\kappa(n)}}$$
where $\kappa(n)$ is a function defined by the largest index of a Lucas number ($L_k$) or twice a Lucas number ($2L_k$) that is congruent to $n \pmod 5$ and bounded above by $n$. Specifically, $\kappa(n) = \max\{\kappa_1(n)-1, \kappa_2(n)\}$, where $\kappa_1$ and $\kappa_2$ correspond to the indices of $L_k$ and $2L_k$ respectively.

• Monotonicity: The function $\sigma_5(n)$ is monotone non-increasing over congruence classes of $n$ modulo 5.
• Jumps: The value of $\sigma_5(n)$ only changes (jumps) when $n$ reaches specific thresholds: $n = 5F_m$, $n = L_m$, or $n = 2L_m$.
• Characterization of Minimizers: The sums achieving this minimum are explicitly characterized. For $n \in [L_m, 2L_{m+1})$, the minimal sums are rotations of $F_m \zeta + F_{m-1} \zeta^3 + F_{m-1} \zeta^4$. For $n \in [2L_m, L_{m+3})$, they are rotations of $F_{m+1} \zeta + F_{m-1} \zeta^2 + F_{m-1} \zeta^3 + F_{m+1} \zeta^4$.

2. Case $n \equiv 0 \pmod 5$ (Theorem 2)
For $n = 5k$, let $\nu(k) = \max \{ m : F_m \le k \}$. The minimal absolute value satisfies:
$$\sigma_5(n)^2 = \frac{\sqrt{5}}{\phi^{2\nu(k)-1}}$$

• Characterization of Minimizers: The minimal sums for $n=5F_m$ are rotations of $2F_m \zeta + F_{m-2} \zeta^2 + F_m \zeta^3 + F_{m+1} \zeta^4$.

Significance
The paper claims that calculating $\sigma_5(n)$ is useful for the study of maximal determinant matrices with entries in the set of 5th roots of unity, as these values appear in determinantal upper bounds. The work extends the understanding of minimal absolute values of roots of unity sums beyond the cases where the minimal polynomial is linear or quadratic ($\ell \in \{2, 3, 4, 6\}$). By reducing the problem to rational approximations of the golden ratio, the authors provide a complete characterization of the minimal values and the specific sums that achieve them, filling a gap in the literature regarding fixed $\ell$ and arbitrary $n$. The results demonstrate that the behavior of these minimal values is deeply connected to the arithmetic properties of Fibonacci and Lucas numbers.