  For $0 \leq \alpha \leq 1/2$, we show an algorithm that does the following.
Given appropriate preprocessing $P(\mathcal{L})$ consisting of $N_\alpha :=
2^{O(n^{1-2\alpha} + \log n)}$ vectors in some lattice $\mathcal{L} \subset
\mathbb{R}^n$ and a target vector $\boldsymbol{t}\in \mathbb{R}^n$, the
algorithm finds $\boldsymbol{y} \in \mathcal{L}$ such that $\|\boldsymbol{y}-
\boldsymbol{t}\| \leq n^{1/2 + \alpha} \eta(\mathcal{L})$ in time
$\mathrm{poly}(n) \cdot N_\alpha$, where $\eta(\mathcal{L})$ is the smoothing
parameter of the lattice.
  The algorithm itself is very simple and was originally studied by
Doulgerakis, Laarhoven, and de Weger (to appear in PQCrypto, 2019), who proved
its correctness under certain reasonable heuristic assumptions on the
preprocessing $P(\mathcal{L})$ and target $\boldsymbol{t}$. Our primary
contribution is a choice of preprocessing that allows us to prove correctness
without any heuristic assumptions.
  Our main motivation for studying this is the recent breakthrough algorithm
for IdealSVP due to Hanrot, Pellet--Mary, and Stehl\'e (to appear in Eurocrypt,
2019), which uses the DLW algorithm as a key subprocedure. In particular, our
result implies that the HPS IdealSVP algorithm can be made to work with fewer
heuristic assumptions.
  Our only technical tool is the discrete Gaussian distribution over
$\mathcal{L}$, and in particular, a lemma showing that the one-dimensional
projections of this distribution behave very similarly to the continuous
Gaussian. This lemma might be of independent interest.
