The Minimal Absolute Value of Sums of Fifth Roots of Unity

Abstract

We determine the minimal absolute value of a non-vanishing sum of $n$ fifth roots of unity chosen with repetition, and characterize the corresponding sums. As a function of $n$, the minimal absolute value is monotone non-increasing over congruence classes of $n$ modulo $5$ and its only jumps occur when $n=5F_m$, $n=L_m$, or $n=2L_m$, where $F_m$ and $L_m$ denote the $m$-th Fibonacci and Lucas numbers respectively. To prove our results we reduce the problem to a series of inequalities involving rational approximations of the golden ratio $φ=(1+\sqrt{5})/2$, the solutions of which can be characterized using the theory of continued fractions.