From \((\Psi V)(t)=\sum _iv_i(1-t)^{m-1-i}\) one reads off, over any commutative ring: (F1) if \(t^j\mid V\) then \(\deg \Psi V{\lt}m-j\); (F2) if \(\deg V{\lt}m-j\) then \((1-t)^j\mid \Psi V\). Let \(d_1+d_2+d_3=m\) and suppose that, over a field, the first \(d_1,d_2,d_3\) digits of \(T^2V\), \(V\), \(TV\) vanish: \(t^{d_1}\mid \Psi ^2V\), \(t^{d_2}\mid V\), \(t^{d_3}\mid \Psi V\). Since \(V=\pm \Psi (\Psi ^2V)\), (F1) gives \(\deg V{\lt}m-d_1=d_2+d_3\); and (F1) applied to \(\Psi V\) gives \(\deg \Psi ^2V{\lt}m-d_3\), whence \((1-t)^{d_3}\mid V\) by (F2). As \(1=t+(1-t)\), the factors \(t^{d_2}\) and \((1-t)^{d_3}\) are coprime, so their product, of degree \(d_2+d_3\), divides \(V\); hence \(V=0\). So every box minor of the frame is nonzero modulo every prime, i.e. \(\pm 1\); the scramble multiplies it by \(\pm 1\) (eq:scr), and Lemma 1 concludes.

For \(\deg F\le N\) put \(\Psi _NF=(1-t)^NF(1/(1-t))\), so \(\Psi =\Psi _{m-1}\), and \(\Psi _{a+b}(FG)=\Psi _aF\cdot \Psi _bG\) whenever \(\deg F\le a\), \(\deg G\le b\) (on binary forms, \(\Psi _N\) is the \(N\)-th symmetric power of the matrix \(g\) of Section 3). Moreover \(\Psi _kt^k=1\), \(\Psi _k(t-1)^k=t^k\), \(\Psi _k(1-t)^k=(-t)^k\), \(\Psi _k1=(1-t)^k\), and for \(\deg F{\lt}k\), with \(\Psi '=\Psi _{k-1}\): \((\Psi '^2F)(0)=F_{k-1}=(\Psi 'F)(1)\). Partial fractions of \(V/(t^k(t-1)^k)\) (over \(\mathbb {Z}\): divide by the monic \(t^k(t-1)^k\), then use a Bézout identity \(1=\alpha t^k+\beta (t-1)^k\) in \(\mathbb {Z}[t]\), obtained by expanding \(1=(t-(t-1))^{2k-1}\)) give

and multiplicativity gives

Write \(\psi _j(W)=(\mathcal K[W]_m)_j=[t^{j}]\, W(1-t)^{-k}\), so that \(d(W)=\mathcal E\, (\psi _j(W))_j\).

Cells and (b). Modulo \(t^k\), \(V\equiv A(t-1)^k\) and \(\Psi V\equiv \Psi 'B\, (1-t)^k\), so the \(y\)-cell determines \(A\) and the \(z\)-cell \(B\) (\(L\) is lower-triangular with diagonal \(\pm 1\), \((t\mp 1)^k\) are units modulo \(t^k\), and \(\Psi '\) is invertible). For \(j{\lt}k\), \(\psi _j(\Psi ^2V)=[t^{j}]\, \Psi '^2Q\), so the \(x\)-cell determines \(Q\), and the three cells determine \(V\). For \(j\ge k\), \(\psi _j(\Psi ^2V)=[t^{j}]\, (-t)^k\bigl(\Psi '^2A\, (1-t)^{-k}+\Psi '^2B\bigr)\) depends on \(A,B\) only; as \(\mathcal E\) changes only rows \(k-2,k-1,k\), this is (b).

(a). From the displays, \(\psi _k(\Psi ^2V)=(-1)^k(A_{k-1}+B_{k-1})\), \(\psi _{k-1}(V)=(-1)^kA_{k-1}\) and, as \(\Psi V(1-t)^{-k}\equiv \Psi 'B\pmod{t^k}\), \(\sum _{j{\lt}k}\psi _j(\Psi V)=(\Psi 'B)(1)=B_{k-1}\). By the definition of \(\mathcal E\) (and \(T^2=\Psi ^2\), \(T=s_k\Psi \)), \(d^x_k=-d^x_{k-1}+(-1)^k(A_{k-1}+B_{k-1})\), \(d^y_{k-1}=-(-1)^kA_{k-1}\) and \(d^z_{k-2}-d^z_{k-1}=s_kB_{k-1}\); with \((-1)^ks_k=-1\) this is (a). Part (c) follows from \(T^3=1\).

Work in cell units, with two points in cells \(n\) and \(n+e\) (toroidally). If some \(|e_i|\ge 2\), the \(i\)-th coordinates differ by at least \(1\ge c(q)\); \(e=0\) is impossible. So let \(e\in \{ -1,0,1\} ^3\setminus \{ 0\} \), and let \(\nu \) be the number of moving axes.

The step. For a moving axis \(d\), \(u_d\) changes by \(e_d\) modulo \(q\), and \(v_d\) changes by \(e_d\) if \(u_d\) wraps (\(\omega _d=1\); this includes the step across the torus boundary, \(n_d\) from \(q^k-1\) to \(0\) or back, where all cell digits wrap; since \(k\ge 2\), \(v_d\) then still changes by exactly \(e_d\)) and is unchanged otherwise (\(\omega _d=0\)). Hence \(r'-r\equiv \sigma \pmod q\) with

\(|\sigma _i|\le 3\), and \(\Delta _i=e_i+((r'_i-r_i)+(f'_i-f_i))/q\), where \(r'_i-r_i\in \{ \sigma _i,\ \sigma _i-q\operatorname {sgn}\sigma _i\} \) (\(\{ 0\} \) if \(\sigma _i=0\)), \(|f'_i-f_i|{\lt}1\), and \(f'_i=f_i\) if only axis \(i\) moves (Proposition 45(b)).

Per-coordinate bound. Call a moving coordinate backward if \(e_i\sigma _i\le 0\) and forward otherwise. For a backward coordinate, \(r'_i-r_i=\sigma _i\) gives \(|\Delta _i|\ge 1-(|\sigma _i|+1)/q\) and \(r'_i-r_i=\sigma _i+qe_i\) gives \(|\Delta _i|\ge 2-(|\sigma _i|+1)/q\); as \(q\ge 4{\gt}|\sigma _i|\), both give \(|\Delta _i|\ge c_{|\sigma _i|+1}\) with \(c_j:=1-j/q\ge 0\). Forward coordinates, and coordinates that do not move, are only bounded by \(0\).

Signs. From 1: (i) If \(e=\pm e_i\) then \(\sigma _i=-e_i\) and there is no jitter, so \(\Delta _i\in \{ e_i(1-1/q),\ e_i(2-1/q)\} \) and \(|\Delta |\ge c_1\). (ii) \(e_i\sigma _i=-1-e_ie_{i+1}+(1-\omega _{i+2})e_ie_{i+2}\) is positive iff \(e_{i+1}=-e_i\), \(e_{i+2}=e_i\) and \(\omega _{i+2}=0\). So a forward coordinate exists only if \(\nu =3\), and it is unique; then \(\sigma _{i+1}=(1-\omega _i)e_i\) (backward, \(|\sigma |\le 1\)) and \(\sigma _{i+2}=-(3-\omega _{i+1})e_i\) (backward, \(|\sigma |\le 3\)). (iii) \(|\sigma _i|=3\) forces \(e_{i+1}=e_i\), \(e_{i+2}=-e_i\), \(\omega _{i+2}=0\), and then \(\sigma _{i+2}=(1-\omega _{i+1})e_i\): backward with \(|\sigma |\le 1\). (iv) If \(\nu =2\) with moving axes \(\{ i,i+1\} \), both are backward by (ii); either \(e_{i+1}=e_i\) and \(|\sigma _{i+1}|\le 1\), or \(e_{i+1}=-e_i\) and \(\sigma _i=0\); the other value is at most \(2\) in absolute value.

Cases. \(\nu =1\): \(|\Delta |\ge c_1\) by (i). \(\nu =2\): \(|\Delta |^2\ge c_2^2+c_3^2\) by (iv). \(\nu =3\) with some backward \(|\sigma _i|=3\): \(|\Delta |^2\ge c_4^2+c_2^2\) by (iii). \(\nu =3\) with a forward coordinate and all backward \(|\sigma |\le 2\): \(|\Delta |^2\ge c_2^2+c_3^2\) by (ii). Otherwise three backward coordinates with \(|\sigma |\le 2\): \(|\Delta |^2\ge 3c_3^2\). As \(c_3\ge c_4\), the minimum is \(c(q)^2\). For \(q\ge 8\), \(c_2^2+c_4^2\ge c_1^2\) (as \(1-10x+19x^2\ge 0\) for \(x=1/q\le 1/8\)) and \(3c_3^2\ge c_1^2\), so \(c(q)=1-1/q\).

Let \(n\ne n'\) be cells and \(D\) a representative of \(X(n')-X(n)\) modulo \(q^3\). Then \(D=q^2E+\Delta \mathbf1\) with an integer vector \(E\equiv n'-n\pmod q\) and \(\Delta =\varphi ([n'])-\varphi ([n])\); every such \(E\) occurs, and the toroidal distance of the pair is the least \(|D|\). With \(\Sigma E=E_x+E_y+E_z\),

The three differences \(E_x-E_y\), \(E_y-E_z\), \(E_z-E_x\) are integers with sum \(0\). If they all vanish, \(E=t\mathbf1\) with \(t\in \mathbb {Z}\); then \(n'\equiv n+t\mathbf1\) lies in the column of \(n\), \(\Delta =0\), \(t\not\equiv 0\pmod q\), and \(|D|^2=3t^2q^4\ge 3q^4\). If one of them has absolute value at least \(2\), the other two sum to its negative, so the three squares sum to at least \(6\) and \(|D|^2\ge 2q^4\). Otherwise they are \(\pm (1,-1,0)\) up to order, i.e. \(E=\pm \mathbf{e}_i+t\mathbf1\) for some \(i\) and \(t\in \mathbb {Z}\). For \(E=\mathbf{e}_i+t\mathbf1\) we get \(D=q^2\mathbf{e}_i+U\mathbf1\) with \(U=\Delta +tq^2\), and

As \(t\) runs over \(\mathbb {Z}\), \(U\) runs over \(\Delta +q^2\mathbb {Z}\), so the least such value is \(\frac23q^4+3\operatorname {dist}(\Delta ,-\frac13q^2+q^2\mathbb {Z})^2\), and \(-\frac13q^2\equiv \frac23q^2\pmod{q^2}\); it is at most \(\frac23q^4+3(\frac12q^2)^2{\lt}2q^4\). A pair with \(E=-\mathbf{e}_i+t\mathbf1\) is, read backwards, a pair with \(E=\mathbf{e}_i-t\mathbf1\). Every pair of neighbouring columns occurs (take any cell \(n\) and \(n'=n+\mathbf{e}_i\)), so the minimum over all pairs is \(\frac23q^4+3g^2\).

(1) The first rows are \(\mathbf{e}_x,\mathbf{e}_y,\mathbf{e}_z\), so \(\mathcal C_q\) is a graph form with the depths \((1,1,1)\), and we use Lemma 2. The box of shape \((1,1,1)\) has determinant \(1\) and puts one point in every cell. A box with depth \(2\) on \(c\), \(1\) on \(c'\) and \(0\) on \(c''\) has, up to sign, the determinant \(a_{c''}\); as \(3a_{c''}=q+w_{c''}\equiv w_{c''}\in \{ \pm 1,\pm 2\} \) and \(\gcd (q,6)=1\), it is a unit. A box with depth \(3\) on \(c\) has, up to sign, the \(2\times 2\) minor of the rows \(a\) and \(b\) on the two columns other than \(c\). Both rows sum to \(0\) modulo \(q\), so in each of them column \(c\) is minus the sum of the other two modulo \(q\), and modulo \(q\) all three such minors equal \(\pm \mu \) with \(\mu =a_xb_y-a_yb_x=a_x+a_y\) (as integers they are \(a_x+a_y\) and \(\pm a_z\)); and \(3\mu =2q+w_x+w_y\equiv -w_z\in \{ -1,-2\} \pmod q\), a unit. Lemma 1 concludes.

(2) We apply Theorem 50. Let \(\bar b_i\in \{ 0,\dots ,q-1\} \) be the residue of \(b\cdot \mathbf{e}_i\), so \(\bar b=(q-1,1,0)\). A step from the cell \(n\) to \(n+\mathbf{e}_i\) changes \(r\) by \(a_i\) and \(j\) by \(\bar b_i\) modulo \(q\), so

The map \([n]\mapsto (r(n),j(n))\) is a bijection from \(\mathbb T\) onto \((\mathbb {Z}/q\mathbb {Z})^2\), because \(\mu \) is a unit; so the two wraps occur independently, in all four combinations (two if \(\bar b_i=0\)). Modulo \(q^2\) the values of \(\Delta _i\) are therefore \(s_i\) and, if \(\bar b_i\ne 0\), \(s_i-q\), where \(s_i=qa_i+\bar b_i\), and \(3\operatorname {dist}(\Delta _i,-\frac13q^2+q^2\mathbb {Z})\) is the least value of \(|3U+q^2|\) over \(U\in s_i+\{ 0,-q\} +q^2\mathbb {Z}\). Here \(3s_i+q^2\) lies between \(2q^2-q+3\) and \(2q^2+2q\), so the least value is attained at \(U=s_i-q^2\) or \(U=s_i-q^2-q\), where \(3U+q^2=-(2q^2-3s_i)\) resp. \(-(2q^2-3s_i+3q)\); every other \(U\) gives at least \(2q^2-4q+3{\gt}q^2+q\). The values \(2q^2-3s_i\) for \(i=x,y,z\) are

all positive, and adding \(3q\) only increases them. So \(3g\) is the least of them, \(q^2-q-3\) resp. \(q^2-2q\), attained only at \(i=y\) (resp. \(i=z\)) with \(U=s_i-q^2\), and Theorem 50 gives \(d^2=\frac23q^4+3g^2\). The closest differences are then \(\pm (q^2\mathbf{e}_i+U\mathbf1)=\pm D\) modulo \(q^3\). The Euclidean minimum distance is at least \(d\), and it is attained without a wrap: \(\varphi ([\mathbf1])=0\) and \(\varphi ([\mathbf{e}_i])=qa_i+\bar b_i=s_i\), so the points of the cells \(\mathbf1\) and \(\mathbf{e}_i\) differ by \(q^2(\mathbf{e}_i-\mathbf1)+s_i\mathbf1=q^2\mathbf{e}_i+(s_i-q^2)\mathbf1=D\) exactly. The expansions follow from \((\delta N^{1/3})^2=\frac23+\frac13(3g/q^2)^2\).

(3) \(3g/q^2=1-\frac1q-\frac3{q^2}\) resp. \(1-\frac2q\) is positive and increases with \(q\). The values at \(q=5\) and \(7\) are \(\sqrt{513}/25\) and \(\sqrt{2009}/49\), and at \(q=37\) (the least \(q\ge 37\) coprime to \(6\), in the class \(q\equiv 1\); the least in the other class is \(41\), with \(0.9913\ldots \)) it is \(0.9823\ldots \).

Let \(2(j+1)\le m\) and choose the depths \(d_a=j+1\), \(d_b=m-j-1\ge j+1\) and \(d_c=0\) for the other coordinates. Prescribing the digits \(0,\dots ,j\) of \(a\) and of \(b\) fixes the coordinate \(a\) of an elementary box of this shape and the first \(j+1\) digits of its coordinate \(b\), and leaves \(q^{m-2(j+1)}\) choices for the other digits of \(b\); by \(t=0\) each such box holds exactly one point. So each prescription is met by \(q^{m-2j-2}\) points, and each value of the pair of digits with index \(j\) by \(q^{2j}\cdot q^{m-2j-2}=q^{m-2}\) points. A map \(f\) as stated would leave the values \((\alpha ,\beta )\), \(\beta \ne f(\alpha )\), unmet.

Let \(n\ne n'\) and let \(D=q^2E+q\, \Delta r\, \mathbf1+\Delta j\) be a representative of \(X(n')-X(n)\) modulo \(q^3\), with an integer vector \(E\equiv n'-n\pmod q\), \(\Delta r\in (-q,q)\) and \(\Delta j\in (-q,q)^3\). If some \(|E_c|\ge 2\), then \(|D_c|\ge 2q^2-q(q-1)-(q-1){\gt}q^2\). So let \(E\in \{ -1,0,1\} ^3\setminus \{ 0\} \) and \(\Sigma E=E_x+E_y+E_z\). The constant \(\beta \) cancels: \(\Delta r\equiv a\cdot E\pmod q\), and \(3a\cdot E=q\Sigma E+w\cdot E\), so \(3\Delta r=w\cdot E+zq\) with an integer \(z\equiv \Sigma E\pmod3\). Hence \(v=E-\frac{z+2\Sigma E}3\mathbf1\) is an integer vector with \(Hv=E+\frac z3\mathbf1\), and

with \(|R_c|\le q\| w\| _1+3(q-1)\), since \(|w\cdot E|\le \| w\| _1\). If \(v\ne 0\), then \(|D|\ge q^2|Hv|-|R|/3\ge q^2-\bigl((\| w\| _1+3)q-3\bigr)/\sqrt3\), as \(H\) is orthogonal. If \(v=0\), then \(E=t\mathbf1\) with \(3t=z+2\Sigma E=z+6t\), so \(z=-3t\) and, as \(w\cdot \mathbf1=0\), \(\Delta r=-tq\); then \(|\Delta r|{\lt}q\) forces \(t=0\), i.e. \(E=0\), a contradiction.

Shifted digits. By Lemma 69, after re-indexing by the cells, the point of cell \(n\) of \(\mathcal C_q\oplus \tau \) has, in coordinate \(c\), the digits \(n_c\), \(r_c(n)\equiv a\cdot n+\beta _c\) and \(j_c(n)\equiv b\cdot n+\beta '_c\) with \(\beta _c=\tau _{c,2}-a\cdot \tau _1\) and \(\beta '_c=\tau _{c,3}-b\cdot \tau _1\), \(\tau _1=(\tau _{x,1},\tau _{y,1},\tau _{z,1})\). So the \(\beta _c\) are congruent if and only if the \(\tau _{c,2}\) are equal, and likewise for \(\beta '_c\) and \(\tau _{c,3}\).

(1) Then \(X(n)=q^2n+\varphi '([n])\mathbf1\) with \(\varphi '=q\, r+j\), \(r=(a\cdot n+\beta )\bmod q\), \(j=(b\cdot n+\beta ')\bmod q\): a column point set. The map \([n]\mapsto (r,j)\) is still a bijection onto \((\mathbb {Z}/q\mathbb {Z})^2\), so in the proof of Theorem 51(2) the same values \(\Delta _i\) occur, and Theorem 50 gives the same \(d\).

(2) Proposition 53 with \(\| w\| _1=4\).

(3) Lower bound. Let \(n\ne n'\) and \(D=q^2E+q\, \Delta r+\Delta j\) a representative, now with \(\Delta r=(\Delta r_c)_c\in (-q,q)^3\). If some \(|E_c|\ge 2\), then \(|D_c|{\gt}q^2\). So let \(E\in \{ -1,0,1\} ^3\setminus \{ 0\} \), and let \(\vartheta \in [0,q)\) be the residue of \(a\cdot E\). The constants \(\beta _c\) cancel: \(\Delta r_c\equiv a\cdot E\) and \(|\Delta r_c|{\lt}q\), so \(\Delta r_c=\vartheta -q\omega _c\) with \(\omega _c\in \{ 0,1\} \) (\(\omega =0\) if \(\vartheta =0\)). Hence \(D=q^2u+\Delta j\) with \(u=E-\omega +\theta \mathbf1\), \(\theta =\vartheta /q\), and \(|D|\ge q^2|u|-\sqrt3(q-1)\). Every \(u_c\) lies in \(\theta +\mathbb {Z}\), so \(|u_c|\ge \| \theta \| \), the distance from \(\theta \) to the nearest integer, and \(\theta \equiv (a\cdot E)/q\pmod1\). If \(\Sigma E=0\), then \(E=\pm (\mathbf{e}_c-\mathbf{e}_d)\), the two moving coordinates of \(u\) are \(\pm (\theta +\{ 0,1\} )\) and \(\pm (\theta -\{ 1,2\} )\), and \(|u|^2\ge \theta ^2+(1-\theta )^2\ge \frac12\). If \(E=\pm \mathbf1\), then \(\vartheta =0\) and \(|u|=\sqrt3\). Otherwise, up to sign, \(E=\mathbf{e}_i\) or \(E=\mathbf1-2\mathbf{e}_i\) or \(E=\mathbf1-\mathbf{e}_i\), with \(a\cdot E\equiv a_i\), \(-2a_i\), \(-a_i\) respectively. As \(3a_i=q+w_i\) with \(0{\lt}a_i{\lt}\frac q2\), \(\| a_i/q\| =\frac{q+w_i}{3q}\) and \(\| 2a_i/q\| =\min \bigl(\frac{q-2w_i}{3q},\frac{2q+2w_i}{3q}\bigr)\); over the entries \(w_i\in \{ -2,1\} \) resp. \(\{ -1,2\} \) the least of these is \(\frac{q-W}{3q}\). So \(|u|\ge \sqrt3\, \| \theta \| \ge (q-W)/(\sqrt3q)\) in these cases, which is also less than \(\frac1{\sqrt2}\) and \(\sqrt3\). Therefore \(|D|\ge q(q-W)/\sqrt3-\sqrt3(q-1)=\bigl(q^2-(W+3)q+3\bigr)/\sqrt3\) for every pair, which is the bound. It increases with \(q\) within each residue class; for \(q\ge 37\) its least value is \(0.4693\ldots \) at \(q=37\) (\(W=4\)), and \(0.5079\ldots \) at \(q=41\) in the class \(W=2\).

Upper bound. Put \(b_c\equiv -\beta _c\pmod q\); they are not all equal. Along \(\mathbf{e}_c\) the shared digit changes by \(a_c\), and \(r_d\) wraps if and only if \(\vartheta =a\cdot n\bmod q\) lies in the cyclic interval \(J^{(c)}_d=\{ b_d-a_c,\dots ,b_d-1\} \) of \(a_c\) consecutive residues of \(\mathbb {Z}/q\mathbb {Z}\). We find \(c\) and a residue \(\vartheta \) that lies in \(J^{(c)}_c\) and in no \(J^{(c)}_d\), \(d\ne c\). If the \(b_d\) are distinct, let \(g_c\ge 1\) be the distance from \(b_c\) forward around \(\mathbb {Z}/q\mathbb {Z}\) to the next of them. The gaps sum to \(q=a_x+a_y+a_z\), so \(g_c\ge a_c\) for some \(c\); then \(\vartheta =b_c-1\) lies in \(J^{(c)}_c\), and it lies in \(J^{(c)}_d\) only if \(b_d\) is at forward distance at most \(a_c-1\) from \(b_c\), which no \(b_d\) is. If exactly two of them are equal, \(b_d=b_{d'}\ne b_c\), then \(J^{(c)}_c\) and \(J^{(c)}_d=J^{(c)}_{d'}\) are two different cyclic intervals of the same length \(a_c{\lt}q\), so neither contains the other. Take a cell \(n\) with \(n_c=0\) and \(a\cdot n\equiv \vartheta \) (possible, as the entries \(a_d\), \(d\ne c\), are units) and \(n'=n+\mathbf{e}_c\), so that the pair does not wrap in the cell digits. Then \(\Delta r_c=a_c-q\) and \(\Delta r_d=a_c\) for \(d\ne c\), so the representative with \(E=\mathbf{e}_c\) is \(D=q^2\mathbf{e}_c+q(a_c\mathbf1-q\mathbf{e}_c)+\Delta j=q\, a_c\mathbf1+\Delta j\), and \(|D|\le \sqrt3\, (q\, a_{\max }+q-1)\).

Every box minor of \(\Phi ^F\) is \(\pm 1\), and each move multiplies every box minor by a unit of \(\mathbb {F}_p\) (Lemma 2; the targets \(\rho \), \(\rho ^2\), \(1\) and the pivots are units). Lemma 1 completes the proof.

The net property is Proposition 56; the hypotheses give \(2\le \rho {\lt}p\), so \(\rho \not\equiv 0\). Let \(v\ne w\) be two points, \(a\ne 0\) the centred difference of their fine positions, and \(M=|a|_\infty \). If \(M\ge p-1\) we are done, since \(p-1\ge \rho ^2-3\). Otherwise let \(\xi _c=\xi _{c,1}\) be the floor changes of Lemma 27(ii) and \(u=(a_x,\ a_y-\xi _z,\ a_z-\xi _x-\xi _y+\xi _z)\). By the displayed relations, \(u\in \Lambda _{\mathcal B^1}\), and \(|u|_\infty \le M+3\). If \(u\ne 0\), Lemma 31(L1) gives \(M+3\ge \rho ^2\). If \(u=0\), then \(a_x=0\), hence \(\xi _x=0\), \(a_y=\xi _z\) and \(a_z=\xi _y-\xi _z\). If \(\xi _z=1\) then \(a_z{\gt}0\), so \(\xi _y\ge 2\), impossible; if \(\xi _z=-1\) then \(a_z{\lt}0\) and \(\xi _y\le -2\), impossible. So \(\xi _z=0\), \(a_y=0\), \(\xi _y=0\) and \(a_z=0\), contradicting \(a\ne 0\). Hence \(M\ge \rho ^2-3\), and Lemma 27(i) with \(\ell =2\) gives \(\delta N^{1/3}\ge (\rho ^2-4)p^{-2/3}\).

Constants. Let \(c_1(p)=(\rho ^2-4)p^{-2/3}\) be the bound with \(\rho \) maximal. Then \(p{\lt}(\rho +1)^2(\rho +2)\), hence

\(G\) increases for \(\rho \ge 3\), since \((\log G)'=9/((\rho -2)(\rho +1))-1/((\rho +1)(\rho +2))\) is positive, and tends to \(1\). Since \(\rho \ge 27\) iff \(p\ge 20412\), and \(G(27)^{1/3}=0.9033\ldots \), the bound exceeds \(0.9\) from this prime on.

Take \(\mathcal C_p\) for \(m=3\) (Theorem 51; for \(p\ge 37\), \(\delta N^{1/3}{\gt}0.9823\), and its covering radius is at most \(\sqrt3N^{-1/3}\) because every cell holds a point), \(\mathcal P_{k,p}\) for \(m=3k\ge 6\), \(\mathcal B^1_{k,p,\rho }\) for \(m=3k+1\ge 4\) and \(\mathcal B^2_{k,p,\rho }\) for \(m=3k+2\ge 5\), with \(\rho \) maximal: \(\rho _1\) and \(\rho _2\), the largest integers with \(\rho _1^2(\rho _1+1)\le p\) and \(\rho _2^3{\lt}p\). For \(p\ge 37\), \(\rho _1\ge 3\) and \(\rho _2\ge 3\). If \(37\le p\le 79\) then \(\rho _1=3\) and the bound of \(\mathcal B^1\) is \(5p^{-2/3}\ge 5\cdot 79^{-2/3}=0.27157\ldots \), the bound of \(\mathcal B^2\) is \(\ge 2\cdot 61^{-1/3}{\gt}0.508\), and \(1-1/p{\gt}0.97\). If \(p\ge 83\) then \(\rho _1\ge 4\) and \(\rho _2\ge 4\), and the bounds exceed \(G(4)^{1/3}=(48/625)^{1/3}{\gt}0.4250\) and \((\rho -1)/(\rho +1)=3/5\) at \(\rho =4\) (Remark 57). All bounds tend to \(1\). For the covering radius: by \(t=0\) the elementary box with depths \(\mathbf k\) (resp. \((k,k,k)\)) around any point of the torus contains a net point, and its diagonal is \(p^{-k}\sqrt{2+p^{-2}}=p^{1/3}\sqrt{2+p^{-2}}\, N^{-1/3}\) for \(m\equiv 1\) and \(p^{-k}\sqrt{1+2p^{-2}}=p^{2/3}\sqrt{1+2p^{-2}}\, N^{-1/3}\) for \(m\equiv 2\). For \(m\equiv 2\), Theorem 35(3) with \(s=3\) gives \(hN^{1/3}\le \sqrt3(\mu +1)p^{-1/3}\) and \(h/\delta \le \sqrt3(\mu +1)/(\rho -1)\), \(\mu =\lceil p/\rho ^2\rceil \); over all primes \(p\ge 11\) the maxima are \(4.2633\ldots {\lt}4.264\) and \(7\sqrt3\), both at \(p=23\) (for \(\rho \ge 4\) use \(\mu {\lt}\rho (1+1/\rho )^3+1\)).