set of integers modulo an integer q in lattice Some literature about lattices set $\mathbb{Z}_{q}$ in $[-\frac{q}{2}, \frac{q}{2})\cap \mathbb{Z}$ but not $[0,q-1]$ such as "lattice signatures without trapdoors" and "lattice based  blind signatures", I don't know the reason. Thank you for explain the reason for me if you know it.
Well, these lattice schemes need to define a 'small vector' $\epsilon$ in such a way that the multiplication of two small vector elements $\epsilon_0 \times \epsilon_1$ is still small; that is, for a random element $A$, we still have $A \approx A + \epsilon_0 \times \epsilon_1$

One way to do this is to define a 'small vector' as one whose elements all have small absolute values; if you define 'small absolute value' tightly enough, then the elements of $\epsilon_0 \times \epsilon_1$ (ignoring the modulus operation) will still probably [1] have terms with absolute values less that $q/k$ (which is 'close enough' - the original schemes had $k=4$).

Of course, you might ask "why don't they define small vectors to have elements with small nonnegative terms?".  Well, if they defined the ring over $x^n+1$ (for example), then something strange happens.  Element $i$ of $A \times B$ can be expressed as

$$A_iB_0 + A_{i-1}B_1 + … + A_0B_i - A_{n-1}B_{i+1} - A_{n-2} B{i+2} - … - A_{i+1} B_{n-1}$$

Specifically, there are $n-i$ terms which are added to the final product, and $i$ terms which are subtracted from the final product.

For small $i$, there are a large number of added terms; if $A, B$ consists of small nonnegative elements, then the result will be strongly weighted toward the positive.  For large $i$ (just a bit smaller than $n$), the result will be strongly weighted toward the negative.  Hence, the result will have a distinct slant, and that's harder to reason with.

In contrast, if with generate $A, B$ with small absolute values (with no bias between positive and negative), no such bias exists; the distribution of the elements of the result is independent of the position within the vector.



[1]: This is where the probability of decryption failure comes in; most systems don't define the error terms small enough that decryption failure can't happen