Table 105 suggests that \(\max _\kappa \lambda _{\mathrm{jit}}(\kappa )p^{-1/s}\to \sqrt{\gamma _s}\), the square root of the Hermite constant (\(1.122\), \(1.189\), \(1.231\), \(1.290\), \(1.345\), \(\sqrt2\) for \(s=3,\dots ,8\)); here \(\lambda _{\mathrm{jit}}(\kappa )\) is the Euclidean minimum of the skeleton lattice \(\Lambda _\kappa \) after one fine cell of jitter is subtracted in each coordinate (Proposition 104). The upper bound holds by the definition of \(\gamma _s\), since \(\lambda _{\mathrm{jit}}\le \lambda _2(\Lambda _\kappa )\le \sqrt{\gamma _s}\, p^{1/s}\). For the lower bound it would suffice that the index-\(p\) sublattices of \(\mathbb {Z}^s\), scaled to covolume \(1\), equidistribute in \(\mathrm{SL}_s(\mathbb {Z})\backslash \mathrm{SL}_s(\mathbb {R})\) as \(p\to \infty \). Indeed, the set of unimodular lattices with \(\lambda _1{\gt}\sqrt{\gamma _s}-\varepsilon \) is open and nonempty, so for large \(p\) it would contain a scaled \(\Lambda _\kappa \) (\(\lambda _1\) is continuous, and the sublattices containing a unit vector have normalized minimum at most \(p^{-1/s}\)), and then \(\lambda _{\mathrm{jit}}\ge \lambda _2-\sqrt s\). Such equidistribution is a theorem of Clozel, Oh and Ullmo on Hecke points [ 7 ] , but as usually stated it is for simply connected \(G\) and Hecke operators \(T_a\) with \(a\in G(\mathbb Q)\), which does not literally contain the index-\(p\) correspondence of \(\mathrm{diag}(p,1,\dots ,1)\notin \mathrm{SL}_s(\mathbb Q)\). The version of Eskin and Oh [ 15 , Thm. 1.2 ] allows \(a\) in the commensurator of the arithmetic group; we have not carried out the reduction in detail, so we do not state the limit as a theorem. The limit would hold at the residue \(m\equiv -1\pmod s\), as \(p\to \infty \) uniformly in \(k\), and would exceed the limit \(\min (\sqrt s/2,\sqrt2)\) of Theorem 119 (at the residue \(m\equiv 0\)) for \(4\le s\le 7\) and equal it at \(s=8\).

\(B(p)\) and \(\rho (p)\) are nondecreasing, so the bounds are positive for every \(p\ge p_0\). For \(s\ge 5\) a rough count gives \(B\approx (p/\sqrt2)^{2/s}\), i.e. \(\delta N^{1/s}\to 2^{-1/s}\) (heuristic). For \(s\ge 9\) the route of Lemma 33(c) needs \(B\) of order \((2s/\pi )^2\), which suggests \(p_0(s)=e^{O(s\log s)}\) (heuristic). Nothing is optimized: the faults cost \(g^*\) cells at small \(p\), and for even \(r\) a block radix, in which each of the two rows of \(A\) is a radix chain on its own group of the full coordinates, has a better constant.