Fine-grained hardness of CVP(P)--- Everything that we can prove (and nothing else)

We show that the Closest Vector Problem in the $\ell_p$ norm ($\mathrm{CVP}_p$) cannot be solved in $2^{(1-\varepsilon)n}$ time for all $p \notin 2\mathbb{Z}$ and $\varepsilon > 0$ (assuming SETH). In fact, we show that the same holds even for (1)~the approximate version of the problem (assuming a gap version of SETH); and (2) $\mathrm{CVP}_p$ with preprocessing, in which we are allowed arbitrary advice about the lattice (assuming a non-uniform version of SETH). For "plain" $\mathrm{CVP}_p$, the same hardness result was shown in [Bennett, Golovnev, and Stephens-Davidowitz FOCS 2017] for all but finitely many $p \notin 2\mathbb{Z}$, where the set of exceptions depended on $\varepsilon$ and was not explicit. For the approximate and preprocessing problems, only very weak bounds were known prior to this work. We also show that the restriction to $p \notin 2\mathbb{Z}$ is in some sense inherent. In particular, we show that no "natural" reduction can rule out even a $2^{3n/4}$-time algorithm for $\mathrm{CVP}_2$ under SETH. For this, we prove that the possible sets of closest lattice vectors to a target in the $\ell_2$ norm have quite rigid structure, which essentially prevents them from being as expressive as $3$-CNFs.