Conditional distribution of an integer error vector, taken from an appropriate discrete Gaussian, given its syndrome [GPV'STOC2008]

Question

I'm reading lemma 5.2 of [GPV'STOC2008, page 18] about conditional distribution of an integer error vector $\mathbf{e}\in\mathbb{Z}^{m}$, taken from an appropriate discrete Gaussian $\mathbf{e}\sim D_{\mathbb{Z}^{m},s}$, given its syndrome $\mathbf{u}=\mathbf{A}\mathbf{e}\mod q$. Below is the image:

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My questions are:

1. Why are they saying the conditional distribution is $\mathbf{t}+D_{\Lambda^{\perp},s,-\mathbf{t}}$? This is an $m$-dimensional vector added to a scaler-value function, not a probability distribution. It's quite confusing to me. Is it a notation for otherthings? Is $\mathbf{t}+D_{\Lambda^{\perp},s,-\mathbf{t}}$ exactly the probability distribution $D_{\Lambda^{\mathbf{u}}_{q}(\mathbf{A})}$?
2. In the last line of the proof, how do they say $\mathbf{v}$ is distributed as $D_{\Lambda^{\perp},s,-\mathbf{t}}$? By writing $\mathbf{e}=\mathbf{t}+\mathbf{e}$, we'll have $D(\mathbf{e})=D(\mathbf{t}+\mathbf{v})=\ldots=D_{\Lambda^{\perp},s,-\mathbf{t}}(\mathbf{e}-\mathbf{t})=D_{\Lambda^{\perp},s,-\mathbf{t}}(\mathbf{v})$.

Answer 1

1. $D_{\Lambda^\perp,s,-\mathbf t}$ is a probability distribution, and the sampling from the conditional distribution that the authors wish to describe is the same as sampling from this distribution and adding the fixed vector $\mathbf t$. A simpler example might be, if I toss six fair coins and wish to describe the deficiency/excess of the number of heads, I could say that it is distributed $-3+\mathrm{Bin}(6,0.5)$.

2. They are just trying to replace the right hand side with a distribution function of a single random variable. This allows them to write $$D(\mathbf e)=D(\mathbf t+\mathbf v)=\mathbf t+D(\mathbf v)=\mathbf t+D_{\Lambda^\perp,s,-\mathbf t}$$ as required.