1 Introduction
Let \(s\ge 3\), let \(q\ge 2\) be an integer all of whose prime factors are at least \(s-1\), and let \(k\ge 1\). Let \(\mathcal P^{U^*_s}_{k,q}\) be the digital net over \(\mathbb {Z}/q\mathbb {Z}\) of Definition 15 for the places \(\infty ,0,1,\dots ,s-2\) and the design \(U^*_s\) of Definition 25, with \(N=q^{sk}\) points. Then:
\(\mathcal P^{U^*_s}_{k,q}\) is a digital \((0,sk,s)\)-net over \(\mathbb {Z}/q\mathbb {Z}\), every cell \(\prod _i[n_iq^{-k},(n_i+1)q^{-k})\) holds exactly one point, and \(h\le \sqrt s\, N^{-1/s}\);
if \(q\ge 5a^*_s+4\), its toroidal (hence also Euclidean) minimum distance satisfies \(\delta \ge (1-a^*_s/q)\, N^{-1/s}\), for every \(k\ge 1\); for \(k\ge 2\) the constant \(a^*_s\) is first-order sharp for this net: \(\delta N^{1/s}=1-a^*_s/q+O(q^{-2})\) as \(q\to \infty \).
Here \(a^*_3=2\), and \(a^*_s\le 2(s-1)^{3/2}(3/2)^{s-2}(s-2)!=e^{s\log s+O(s)}\) for \(s\ge 4\).
Reachability. By Lemma 22(iii) every column \(E_{d,j}\) is admissible at carry depth \(j\), with polynomial \(g_{d,j}\in R_S[X]\) of degree \(j\) and leading coefficient \(\gamma _j\), an \(S\)-unit; so by Proposition 18 the off-diagonal part of \(U^*_s\) is reachable with \(\lambda '\equiv 1\) over \(R'=R_S\), and the diagonal is free (Lemma 2(a)).
(1) Every prime factor of \(q\) is at least \(s-1{\gt}\nu \), so \(\gcd (q,\nu !)=1\), and Corollary 16 gives a digital \((0,sk,s)\)-net with one point in every cell, for every \(k\ge 1\). A cell has diameter \(\sqrt s\, q^{-k}=\sqrt s\, N^{-1/s}\), which bounds \(h\).
(2) Lower bound. (H1) and (H2) of Lemma 4 hold by Proposition 14 and Definition 15, with \(L=\min (k,s-1)\). \(U^*_s\) is diagonally dominant by construction, with the maximum over all carry depths, so Proposition 21 gives \(\delta q^k\ge 1-a^*_s/q\) for \(q\ge 5a^*_s+4\) and every \(k\ge 1\).
Sharpness (\(k\ge 2\)). Let \(i\) attain \(a^*_s=\alpha _i\). For \(q{\gt}2a^*_s\) the entries of \(F_0\) are centred, so \(\tilde M=F_0\). By (H2) the own-axis step \(e_i\) from a cell whose last digit of \(n_i\) is not \(q-1\) changes coordinate \(i\) by exactly \(1-a^*_s/q\) cells if its first offset digit is at least \(a^*_s\) (no wrap, no jitter), and every other coordinate \(u\) by less than \((|\tilde M_{ui}|+1)/q\) if \(r_u\) avoids the \(|\tilde M_{ui}|\) values from which adding \(\tilde M_{ui}\) wraps. For each \(u\ne i\) the \(t=0\) box with depths \(k+1\) on \(i\) and \(u\) and \(0\) elsewhere (total \(2k+2\le sk\)) makes \((n_i,r_i,n_u,r_u)\) jointly uniform (that box fixes exactly these \(2k+2\) digits, so every value of them occurs equally often), so among the cells satisfying the conditions on \(n_i\) and \(r_i\) the fraction with a wrapping \(r_u\) is \(|\tilde M_{ui}|/q\); for \(q{\gt}\sum _{u\ne i}|\tilde M_{ui}|\) a union bound leaves a cell. Its pair of points lies inside the unit cube, so
The bound on \(a^*_s\). Let \(s\ge 4\). By Lemma 24 it suffices to show \(\eta _{d,j}\le \Xi _s:=2\sqrt{s-1}\, (s-2)!\) for every column \(d\) and carry depth \(j\); then every entry of \(U^*_s\) is at most \(\max _{d,j}\| E_{d,j}\| _2\le (3/2)^{s-2}\Xi _s\), and \(a^*_s\le (s-1)(3/2)^{s-2}\Xi _s=2(s-1)^{3/2}(3/2)^{s-2}(s-2)!\). By Lemma 23 it suffices to give, at each place of \(\mathbb Q\), one monic \(g\) of degree \(j\).
Finite column \(d\). Archimedean place: all nodes \(y_c\) lie in \([-1,1]\) and \(|w_c|\le 1\), so the monic Chebyshev polynomial \(2^{1-j}T_j\) (\(j\ge 1\); \(g=1\) for \(j=0\)) gives \(A_{j,\mathbb {R}}\le 2\sqrt{s-1}\). At \(p\in S\): let \(\mathcal R_p\) be the finite rows \(c\ne d\) with \(p\mid c-d\), \(e_c=v_p(c-d)\ge 1\) for \(c\in \mathcal R_p\), let \(\mathcal R'\subset \mathcal R_p\) consist of \(\min (j,|\mathcal R_p|)\) rows with the largest \(e_c\), and \(g_p=\prod _{r\in \mathcal R'}(X-y_r)\cdot X^{j-|\mathcal R'|}\). For \(c\in \mathcal R'\) the entry vanishes. For \(c,r\in \mathcal R_p\), \(|y_c-y_r|_p=|(r-c)/((c-d)(r-d))|_p\le p^{\max (e_c,e_r)}\); so for \(c\in \mathcal R_p\setminus \mathcal R'\) (then \(|\mathcal R'|=j\) and \(e_c\le e_r\) for \(r\in \mathcal R'\)) the entry has \(p\)-adic size at most \(p^{e_c+\sum _{r\in \mathcal R'}e_r}\le p^{\sum _{r\in \mathcal R_p}e_r}\). For a finite row \(c\notin \mathcal R_p\), and for the row \(\infty \) (\(y=0\), \(w=1\)), \(|w_c|_p=1\), \(|y_c|_p\le 1\) and \(|y_c-y_r|_p\le p^{e_r}\), so the size is at most \(p^{\sum _{r\in \mathcal R'}e_r}\). Hence \(A_{j,p}\le p^{\sum _{r\in \mathcal R_p}e_r}=p^{v_p(d!\, (\nu -d)!)}\), and \(\prod _{p\in S}A_{j,p}\le d!\, (\nu -d)!\le \nu !\), since every prime factor of \(d!(\nu -d)!\) lies in \(S\). So \(\eta _{d,j}\le 2\sqrt{s-1}\, \nu !\).
Column \(\infty \). The nodes are \(y_c=c\in \{ 0,\dots ,\nu \} \) with \(w_c=1\). At the archimedean place the monic Chebyshev polynomial of \([0,\nu ]\) gives \(A_{j,\mathbb {R}}\le 2\sqrt{s-1}\, (\nu /4)^j\). At \(p\in S\) the falling factorial \(X(X-1)\cdots (X-j+1)\) takes the values \(j!\binom cj\), so \(A_{j,p}\le |j!|_p\) and \(\prod _pA_{j,p}\le 1/j!\). So \(\eta _{\infty ,j}\le 2\sqrt{s-1}\, (\nu /4)^j/j!\le 2\sqrt{s-1}\, e^{\nu /4}\le 2\sqrt{s-1}\, \nu !\) for \(\nu \ge 2\).
Finally, with \(\nu =s-2\) the elementary bounds \(\nu ^\nu e^{-\nu }\le \nu !\le \nu ^\nu \) give \(\bigl|\log \bigl(2(s-1)^{3/2}(3/2)^{s-2}(s-2)!\bigr)-s\log s\bigr|\le 3s\) for \(s\ge 4\), so the bound is \(e^{s\log s+O(s)}\); and \(a^*_3=2\) is the display before Table tab:astar. (At the archimedean place the Chebyshev polynomial even gives \(A_{j,\mathbb {R}}\le \sqrt{s-1}\) for a finite column, since \(|2^{1-j}T_j|\le 1\) on \([-1,1]\) for \(j\ge 1\); the factor \(2\) in \(\Xi _s\) is slack.)