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We show improved fine-grained hardness of two key lattice problems in the$\ell_p$ norm: Bounded Distance Decoding to within an $\alpha$ factor of theminimum distance ($\mathrm{BDD}_{p, \alpha}$) and the (decisional)$\gamma$-approximate Shortest Vector Problem ($\mathrm{SVP}_{p,\gamma}$),assuming variants of the Gap (Strong) Exponential Time Hypothesis (Gap-(S)ETH).Specifically, we show:  1. For all $p \in [1, \infty)$, there is no $2^{o(n)}$-time algorithm for$\mathrm{BDD}_{p, \alpha}$ for any constant $\alpha > \alpha_\mathsf{kn}$,where $\alpha_\mathsf{kn} = 2^{-c_\mathsf{kn}} < 0.98491$ and $c_\mathsf{kn}$is the $\ell_2$ kissing-number constant, unless non-uniform Gap-ETH is false.  2. For all $p \in [1, \infty)$, there is no $2^{o(n)}$-time algorithm for$\mathrm{BDD}_{p, \alpha}$ for any constant $\alpha > \alpha^\ddagger_p$, where$\alpha^\ddagger_p$ is explicit and satisfies $\alpha^\ddagger_p = 1$ for $1\leq p \leq 2$, $\alpha^\ddagger_p < 1$ for all $p > 2$, and $\alpha^\ddagger_p\to 1/2$ as $p \to \infty$, unless randomized Gap-ETH is false.  3. For all $p \in [1, \infty) \setminus 2 \mathbb{Z}$ and all $C > 1$, thereis no $2^{n/C}$-time algorithm for $\mathrm{BDD}_{p, \alpha}$ for any constant$\alpha > \alpha^\dagger_{p, C}$, where $\alpha^\dagger_{p, C}$ is explicit andsatisfies $\alpha^\dagger_{p, C} \to 1$ as $C \to \infty$ for any fixed $p \in[1, \infty)$, unless non-uniform Gap-SETH is false.  4. For all $p > p_0 \approx 2.1397$, $p \notin 2\mathbb{Z}$, and all $C >C_p$, there is no $2^{n/C}$-time algorithm for $\mathrm{SVP}_{p, \gamma}$ forsome constant $\gamma > 1$, where $C_p > 1$ is explicit and satisfies $C_p \to1$ as $p \to \infty$, unless randomized Gap-SETH is false.

Improved Hardness of BDD and SVP Under Gap-(S)ETH