Shouldn't this happen as long as $\mathbf{a}$ and $\mathbf{b}$ are linearly dependent mod $p$? For if so, you are talking about points in $\mathbb{Z}^n$ which reduce mod $p$ in two some $n-2$ dimensional subspace of $\mathbb{F}_p^n$. After choosing a basis, this amounts to choosing $p^{n-2}$ coefficients. Thus the lattice contains $p^{n-2}$ points in any box $p\times\stackrel{n}{\cdots}\times p$ box.

Maybe I am missing something...

Shouldn't this happen as long as $\mathbf{a}$ and $\mathbf{b}$ are linearly dependent mod $p$? For if so, you are talking about points in $\mathbb{Z}^n$ which reduce mod $p$ in two some $n-2$ dimensional subspace of $\mathbb{F}_p^n$. After choosing a basis, this amounts to choosing $p^{n-2}$ coefficients. Thus the lattice contains $p^{n-2}$ points in any box $p\times\stackrel{n}{\cdots}\times p$ box.