In MITM attacks against the NTRU cryptosystem, we exploit the fact that in the ring of truncated polynomials of degree $n-1$ it holds that $$fg=h\mod q$$ for our secret and public keys $f,h$. The basic idea is splitting $f$ into $f_1,f_2$ such that $f_1+f_2=f$ and therefore considering $$f_1h=g-f_2h. $$This is almost like finding a collision in the function $f(x)=xh$, if it weren't for the presence of $g$. So we must introduce an auxiliary function $a(x)$, which according to the notes I'm reading is defined as follows:
To search for near-collisions an auxiliary function $a(x)$ is needed. This function takes a vector of length n and in each coordinate $x_i$ returns $\mathbb I (x_i > 0)$. If $g$ does not cause the coordinates of $−f_2 · h$ to change sign, i.e. $a(−f_2 · h) \ne a(−f_2 · h + g)$, we have that $a(f_1 h) = a(−f_2 h)$.
I don't quite understand what this function is supposed to do. Can anyone explain it to me in simple terms?