Suppose $x_1,x_2,...,x_b$ is a Complete Residue System (CRS) mod $b$.
is there a CRS say $y_1,y_2,...,y_b$ s.t. $\vert y_i \vert \leq \frac{b}{2}$ $\forall i = 1,2,...,b$
I want to say you CAN find one, namely, does the following work for mod $4$:
take $x_1,x_2,x_3,x_4$ be $\{4,9,14,19\}$
then $4 \in [0]_4$, $9 \in [1]_4$, $14 \in [2]_4$, $19 \in [3]_4$
I claim $y_1,y_2,...,y_b$ is $\{0,1,2,-1\}$
which is still a CRS, has 4 elements AND
$0 \in [0]_4$, $1 \in [1]_4$, $2 \in [2]_4$, $-1 \in [3]_4$
Making $y_1,y_2,...,y_b$ = $\{0,1,2,-1\}$ a CRS modulo 4.
Thanks in advance!!!