Minimum Value of Complex Expression with Cube Root of Unity

To find the minimum value of the expression $|a + b\omega + c\omega^2|^2$ for distinct non-zero integers $a, b, c$, we first simplify the modulus using the properties of the complex cube root of unity $\omega$.

Since $\omega = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$ and $\omega^2 = -\frac{1}{2} - i\frac{\sqrt{3}}{2}$, we can expand the squared magnitude as follows: $|a + b\omega + c\omega^2|^2 = (a + b\omega + c\omega^2)(a + b\omega^2 + c\omega)$ $= a^2 + b^2 + c^2 - ab - bc - ca$ $= \frac{1}{2} \left[ (a - b)^2 + (b - c)^2 + (c - a)^2 \right]$

Since $a, b, c$ are distinct non-zero integers, without loss of generality, assume $a < b < c$. The minimal possible absolute differences between consecutive distinct integers are $b - a \ge 1$ and $c - b \ge 1$, which implies $c - a \ge 2$.

Substituting the minimum possible differences $b - a = 1$, $c - b = 1$, and $c - a = 2$ (which is satisfied by distinct non-zero integers such as $a = 1, b = 2, c = 3$), we obtain: $\text{Minimum value} = \frac{1}{2} \left[ 1^2 + 1^2 + 2^2 \right] = \frac{1}{2} (1 + 1 + 4) = 3$

Thus, the minimum value of the set is $3$.