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Specialized external inputs

For finite disjoint sets of places $S,T$ of $\mathbb Q$, with $\infty\in S$ and with $T$ consisting of odd primes, let $G_T^S$ be the Galois group of the maximal pro-$2$ extension of $\mathbb Q$ that is unramified outside $T$ and in which every place of $S$ splits completely. Write $d(G)$ and $r(G)$ for the generator and relation ranks of a pro-$2$ group. The proof uses the following external statements in precisely these specialized forms.

1. The Frattini quotient of $G_T^{\{\infty\}}$ corresponds to the maximal totally real multiquadratic extension $L_T$ unramified outside $T$. Quadratic theory gives

$$d(G_T^{\{\infty\}})= \begin{cases} |T|-1,&\text{if some prime in }T\text{ is }3\pmod4,\\ |T|,&\text{otherwise.} \end{cases} \tag{1}$$

The companion also points to Koch, Galois theory of $p$-extensions, Springer Monographs in Mathematics (2002), Theorem 11.8. 2. If each finite prime in $S$ splits completely in $L_T(i)$, imposing those splitting conditions adds $|S|-1$ Frobenius relations. Those elements are already trivial in the Frattini quotient, and the Shafarevich relation-rank bound, in the form cited by the companion to Koch, Theorems 11.5 and 11.8, gives

$$d(G_T^S)=d(G_T^{\{\infty\}}),\qquad r(G_T^S)\leq d(G_T^{\{\infty\}})+|S|-1. \tag{2}$$

The underlying papers cited there are Igor R. Shafarevich, Extensions à points de ramification donnés (Russian), Publications Mathématiques de l'IHÉS 18 (1963), 71--92, and its English translation, Extensions with given points of ramification, AMS Translations, Series 2 59 (1966), 128--149. 3. The Golod--Shafarevich theorem implies that a finitely generated pro-$2$ group with $r(G)\leq d(G)^2/4$ is infinite. The cited source is E. S. Golod and I. R. Shafarevich, On the class field tower, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 261--272; English translation, AMS Translations (2) 48 (1965), 91--102. 4. Every finite layer $L$ is totally real and tamely ramified only over $T$, so

$$|\operatorname{Disc}L| \leq\prod_{q\in T}q^{[L:\mathbb Q]}.$$

For $K=L(i)$, the companion uses

$$|\operatorname{Disc}K| \leq\prod_{q\in T\cup\{2\}}q^{2[L:\mathbb Q]}. \tag{3}$$

5. For $[K:\mathbb Q]\geq4$, the proof uses $h(K)\leq|\operatorname{Disc}K|$. Its stated reference is Armand Borel and Gopal Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publications Mathématiques de l'IHÉS 69 (1989), 119--171, p. 143, equation (7).
6. If $K$ is totally imaginary of degree $2f$, the covolume of $\mathcal O_K$ under the Minkowski embedding into $\mathbb C^f$ is

$$2^{-f}\sqrt{|\operatorname{Disc}K|}. \tag{4}$$

These exact specializations and their applicability are part of the present chain. Their external proofs were not reconstructed or checked against separately retained primary PDFs.

An explicit infinite tower

Take

$$T=\{3,5,7,11,13,17\},\qquad S=\{101,\infty\}.$$

The multiquadratic Frattini field may be written as

$$L_T=\mathbb Q(\sqrt5,\sqrt{13},\sqrt{17},\sqrt{21},\sqrt{33}). \tag{5}$$

The five displayed square classes are independent, so $[L_T:\mathbb Q]=32$. The prime $101$ splits completely in $L_T(i)$. The assertion can be checked directly from

$$45^2\equiv5,\quad35^2\equiv13,\quad44^2\equiv17,\quad 18^2\equiv21,\quad29^2\equiv33,\quad10^2\equiv-1\pmod{101}. \tag{6}$$

Since $T$ contains primes congruent to $3$ modulo $4$, (1) gives $d=5$. Equation (2) gives $r\leq6$, and

$$6<\frac{5^2}{4}.$$

The Golod--Shafarevich criterion therefore makes $G_T^S$ infinite. Its finite layers supply totally real fields $L_j$ with $f_j=[L_j:\mathbb Q]\to\infty$, unramified outside $T$, such that $101$ splits completely in $L_j(i)$. Put

$$K_j=L_j(i),\qquad r=\prod_{q\in T\cup\{2\}}q =2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17=510510. \tag{7}$$

Then $K_j$ is a CM field of degree $2f_j$, and (3) gives

$$|\operatorname{Disc}K_j|\leq r^{2f_j}. \tag{8}$$

Norm-one elements and lattice parameters

Set $p=101$ and

$$k=\left\lceil\frac{18r^3}{\pi}\right\rceil-1. \tag{9}$$

Because $p$ splits completely in $K_j$, its $2f_j$ primes form $f_j$ conjugate pairs. Apply [[discrete_geometry/alon_2026_remarks_disproof_unit_distance_conjecture/lemma_2_2_norm_one_elements|Lemma 2.2]] to one prime from each pair, with every exponent equal to $k$. The ideal $\mathfrak Q$ and its denominator become

$$\mathfrak Q=p^k\mathcal O_{K_j},\qquad D=p^{2k}. \tag{10}$$

After discarding finitely many layers if needed so that $[K_j:\mathbb Q] \geq4$, (8) and the class-number input give

$$|U|\geq\frac{(k+1)^{f_j}}{h(K_j)} \geq\left((k+1)r^{-2}\right)^{f_j}. \tag{11}$$

Let

$$\Lambda_j=p^{-2k}\mathcal O_{K_j}\subset\mathbb C^{f_j},\qquad \delta=p^{-2k},\qquad u=(k+1)r^{-2}. \tag{12}$$

Here the embedding uses one member of every conjugate pair of complex embeddings. For nonzero $a\in\mathcal O_{K_j}$, the integer norm is nonzero, so some complex embedding has $|\sigma(a)|\geq1$. Thus each nonzero element of $\Lambda_j$ has some coordinate of magnitude at least $\delta$. Every coordinate projection is a field embedding and hence is injective on the lattice. Because $K_j$ is CM, every element counted in (11) has magnitude one in every coordinate.

By (4), scaling in all $f_j$ complex coordinates gives

$$\operatorname{covol}(\Lambda_j) =2^{-f_j}\delta^{2f_j}\sqrt{|\operatorname{Disc}K_j|}.$$

Consequently

$$\delta^{-2}\operatorname{covol}(\Lambda_j)^{1/f_j} =\frac12|\operatorname{Disc}K_j|^{1/(2f_j)} \leq\frac r2.$$

Take $v=r/2$. The choice (9) gives

$$u=\left\lceil\frac{18r^3}{\pi}\right\rceil r^{-2} >\frac{18r}{\pi}=\frac{36v}{\pi}. \tag{13}$$

Thus $u,v,\delta$ are fixed as $f_j\to\infty$ and satisfy every hypothesis of Lemma 2.1.

Source and proof scope

The tower inputs and proof of Theorem 1.1 are on pp. 4--6 of the retained arXiv v1 manuscript. The same-paper choices, splitting check, ideal application, discriminant and covolume calculations are all included above. The six numbered external inputs are used as stated; this page does not claim their proofs.

Used by. [[discrete_geometry/alon_2026_remarks_disproof_unit_distance_conjecture/theorem_1_1_e90_e92|Theorem 1.1]].