A sharp upper bound for the 2‐torsion of class groups of multiquadratic fields

MathematikaVolume 68, Issue 1 p. 237-258 RESEARCH ARTICLEOpen Access A sharp upper bound for the 2-torsion of class groups of multiquadratic fields Peter Koymans, Corresponding Author Peter Koymans koymans@mpim-bonn.mpg.de Max Planck Institute for Mathematics, Bonn, Germany Correspondence Peter Koymans, Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany. Email: koymans@mpim-bonn.mpg.deSearch for more papers by this authorCarlo Pagano, Carlo Pagano Max Planck Institute for Mathematics, Bonn, GermanySearch for more papers by this author Peter Koymans, Corresponding Author Peter Koymans koymans@mpim-bonn.mpg.de Max Planck Institute for Mathematics, Bonn, Germany Correspondence Peter Koymans, Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany. Email: koymans@mpim-bonn.mpg.deSearch for more papers by this authorCarlo Pagano, Carlo Pagano Max Planck Institute for Mathematics, Bonn, GermanySearch for more papers by this author First published: 29 March 2022 https://doi.org/10.1112/mtk.12123AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract Let K be a multiquadratic extension of Q $\mathbb {Q}$ and let Cl + ( K ) $\text{Cl}^{+}(K)$ be its narrow class group. Recently, the authors (Koymans and Pagano, Int. Math. Res. Not. 2022 (2022), no. 4, 2772–2823) gave a bound for | Cl + ( K ) [ 2 ] | $|\text{Cl}^{+}(K)[2]|$ only in terms of the degree of K and the number of ramifying primes. In the present work we show that this bound is sharp in a wide number of cases. Furthermore, we extend this to ray class groups. 1 INTRODUCTION The class group is one of the most fundamental invariants of a number field K. Providing non-trivial upper bounds for the l-torsion of class groups in terms of the discriminant Δ K / Q $\Delta _{K/\mathbb {Q}}$ of a general number field K has been an active area of research with connections to elliptic curves and diophantine approximation [1-3, 6, 9-11, 14, 15]. For extensions K / Q $K/\mathbb {Q}$ of degree a power of a prime l much more is known. For instance, for l = 2 $l=2$ and K / Q $K/\mathbb {Q}$ a quadratic extension, Gauss [4] showed that dim F 2 Cl + ( K ) [ 2 ] = ω ( Δ K / Q ) − 1 . \begin{equation*} \text{dim}_{\mathbb {F}_2} \text{Cl}^{+}(K)[2] = \omega (\Delta _{K/\mathbb {Q}}) - 1. \end{equation*} Here Cl + ( K ) $\text{Cl}^{+}(K)$ denotes the narrow class group of the field K and ω ( a ) $\omega (a)$ denotes the number of prime factors of a non-zero integer a. Recently, the authors [7] generalized Gauss' result to multiquadratic fields. More specifically, we obtained the following result, which is Theorem 1.1 of [7]. Call a vector ( a 1 , ⋯ , a n ) ∈ Z ⩾ 2 n $(a_1, \dots , a_n) \in \mathbb {Z}_{\geqslant 2}^n$ acceptable if the a i $a_i$ are squarefree, pairwise coprime and only have prime factors congruent to 1 modulo 4. Theorem 1.1.Let n be a positive integer and let ( a 1 , ⋯ , a n ) ∈ Z ⩾ 2 n $(a_1, \dots , a_n) \in \mathbb {Z}_{\geqslant 2}^n$ be acceptable. Then we have dim F 2 Cl + ( Q ( a 1 , ⋯ , a n ) ) [ 2 ] ⩽ ω ( a 1 · … · a n ) · 2 n − 1 − 2 n + 1 . \begin{equation*} \mbox{dim}_{\mathbb {F}_2} \mbox{Cl}^{+}(\mathbb {Q}(\sqrt {a_1}, \dots , \sqrt {a_n}))[2] \leqslant \omega (a_1 \cdot \ldots \cdot a_n) \cdot 2^{n - 1} - 2^n + 1. \end{equation*} A similar upper bound has subsequently been established by Klüners and Wang in [5, Theorem 2.1] for extensions K / Q $K/\mathbb {Q}$ of degree a power of l. However, when specialized to the multiquadratic fields considered above, their bound in the worst-case scenario is twice as large as the one in Theorem 1.1. This work is devoted to showing that the bound in Theorem 1.1 is sharp for every n ∈ Z ⩾ 1 $n \in \mathbb {Z}_{\geqslant 1}$ . An acceptable vector ( a 1 , ⋯ , a n ) $(a_1, \dots , a_n)$ is said to be maximal in case the inequality of Theorem 1.1 is an equality. Among other things, we have given a recursive characterization of maximal vectors (see [7, Theorem 1.2]), which we reproduce here. Write π S $\pi _S$ for the projection on the coordinates in S, write H 2 + ( K ) $H_2^+(K)$ for the maximal multiquadratic unramified (at all finite places) extension of K, and write [ n ] : = { 1 , ⋯ , n } $[n] := \lbrace 1, \dots , n\rbrace$ . Theorem 1.2.Let n be a positive integer and let ( a 1 , ⋯ , a n ) $(a_1, \dots , a_n)$ be an acceptable vector. Then the following are equivalent. (a) The vector ( a 1 , ⋯ , a n ) $(a_1, \dots , a_n)$ is maximal, that is, dim F 2 Cl + ( Q ( a 1 , ⋯ , a n ) ) [ 2 ] = ω ( a 1 · … · a n ) · 2 n − 1 − 2 n + 1 . \begin{equation*} \mbox{dim}_{\mathbb {F}_2} \mbox{Cl}^{+}(\mathbb {Q}(\sqrt {a_1}, \dots , \sqrt {a_n}))[2] = \omega (a_1 \cdot \ldots \cdot a_n) \cdot 2^{n - 1}-2^n+1. \end{equation*} (b) For every j ∈ [ n ] $j \in [n]$ , the vector π [ n ] − { j } ( a 1 , ⋯ , a n ) $\pi _{[n] - \lbrace j\rbrace }(a_1, \dots , a_n)$ is maximal and every prime divisor p of a j $a_j$ splits completely in H 2 + ( Q ( { a m } m ∈ [ n ] − { j } ) ) $H_2^{+}(\mathbb {Q}(\lbrace \sqrt {a_m}\rbrace _{m \in [n]-\lbrace j\rbrace }))$ . (c) For every j ∈ [ n ] $j \in [n]$ , the vector π [ n ] − { j } ( a 1 , ⋯ , a n ) $\pi _{[n] - \lbrace j\rbrace }(a_1, \dots , a_n)$ is maximal and for every prime divisor p of a j $a_j$ , one (or equivalently any) prime above p in the field Q ( { a m } m ∈ [ n ] − { j } ) $\mathbb {Q}(\lbrace \sqrt {a_m}\rbrace _{m \in [n]-\lbrace j\rbrace })$ belongs to 2 Cl + ( Q ( { a m } m ∈ [ n ] − { j } ) ) $2\mbox{Cl}^{+}(\mathbb {Q}(\lbrace \sqrt {a_m}\rbrace _{m \in [n]-\lbrace j\rbrace }))$ . In particular, Theorem 1.2 recovers the equality of Gauss' theorem for n = 1 $n=1$ as a special case. It is then natural to ask whether for every positive integer n, one can find maximal vectors of dimension n. As the reader can sense from the characterization given in Theorem 1.2, it is not at all obvious how to do this. A naive inductive approach based on the Chebotarev density theorem runs into severe difficulties, since one needs to simultaneously guarantee splitting of a prime p in a field K q $K_q$ depending on q and of q in a field K p $K_p$ depending on p. To circumvent this problem, we use combinatorial ideas from [12], which we explain here from first principles in order to make the present work self-contained (see Section 2). Our main theorem shows that one can find maximal vectors ( a 1 , … , a n ) $(a_1, \ldots , a_n)$ for every n. Moreover, for any fixed n, we show that Theorem 1.1 is sharp for a wide number of choices of ( ω ( a 1 ) , ⋯ , ω ( a n ) ) $(\omega (a_1), \dots , \omega (a_n))$ . More precisely, we establish the following. Theorem 1.3. (a) Take n ∈ Z > 3 $n \in \mathbb {Z}_{> 3}$ and take ( k 1 , ⋯ , k n ) ∈ Z ⩾ 1 × ( 2 · Z ⩾ 1 ) n − 1 $(k_1, \dots , k_n) \in \mathbb {Z}_{\geqslant 1} \times (2 \cdot \mathbb {Z}_{\geqslant 1})^{n-1}$ . Then there are infinitely many acceptable vectors ( a 1 , … , a n ) $(a_1, \ldots , a_n)$ with ω ( a i ) = k i $\omega (a_i) = k_i$ for each i ∈ [ n ] $i \in [n]$ and dim F 2 Cl + ( Q ( a 1 , ⋯ , a n ) ) [ 2 ] = ω ( a 1 · … · a n ) · 2 n − 1 − 2 n + 1 . \begin{equation*} \mbox{dim}_{\mathbb {F}_2} \mbox{Cl}^{+}(\mathbb {Q}(\sqrt {a_1}, \dots , \sqrt {a_n}))[2] = \omega (a_1 \cdot \ldots \cdot a_n) \cdot 2^{n - 1} - 2^n + 1. \end{equation*} (b) Take ( k 1 , k 2 , k 3 ) ∈ Z ⩾ 1 3 $(k_1, k_2, k_3) \in \mathbb {Z}_{\geqslant 1}^3$ . Then there are infinitely many acceptable vectors ( a 1 , a 2 , a 3 ) $(a_1, a_2, a_3)$ with ω ( a i ) = k i $\omega (a_i) = k_i$ for each i ∈ { 1 , 2 , 3 } $i \in \lbrace 1, 2, 3\rbrace$ and dim F 2 Cl + ( Q ( a 1 , a 2 , a 3 ) ) [ 2 ] = ω ( a 1 a 2 a 3 ) · 4 − 7 . \begin{equation*} \mbox{dim}_{\mathbb {F}_2} \mbox{Cl}^{+}(\mathbb {Q}(\sqrt {a_1}, \sqrt {a_2}, \sqrt {a_3}))[2] = \omega (a_1a_2a_3) \cdot 4 - 7. \end{equation*} We speculate that the condition ( k 1 , ⋯ , k n ) ∈ Z ⩾ 1 × ( 2 · Z ⩾ 1 ) n − 1 $(k_1, \dots , k_n) \in \mathbb {Z}_{\geqslant 1} \times (2 \cdot \mathbb {Z}_{\geqslant 1})^{n-1}$ can also be removed for n > 3 $n>3$ , but this seems to be out of reach with the techniques employed in this work. We next turn our attention to ray class groups. First of all, let us notice that the 2-torsion of the ordinary class group of a number field K cannot be larger than the 2-torsion of the narrow class group of K. Hence the upper bound in Theorem 1.1 is also an upper bound for | Cl ( Q ( a 1 , ⋯ , a n ) ) [ 2 ] | $|\text{Cl}(\mathbb {Q}(\sqrt {a_1}, \dots , \sqrt {a_n}))[2]|$ . Less obvious is whether also this bound is sharp. Similarly, fix an integer c, which we take in this paper to be a squarefree product of primes congruent to 1 modulo 4 (see the end of this introduction for some motivation on this assumption). Recall that the ray class group Cl ( Q ( a 1 , ⋯ , a n ) , c ) $\text{Cl}(\mathbb {Q}(\sqrt {a_1}, \dots , \sqrt {a_n}),c)$ is by definition the quotient of the free abelian group on the set of prime ideals of Q ( a 1 , ⋯ , a n ) $\mathbb {Q}(\sqrt {a_1}, \dots , \sqrt {a_n})$ coprime to c by the group of principal fractional ideals that admit a generator α congruent to 1 modulo c. From the definition one sees that Cl ( Q ( a 1 , ⋯ , a n ) , c ) $\text{Cl}(\mathbb {Q}(\sqrt {a_1}, \dots , \sqrt {a_n}),c)$ fits into an exact sequence 0 → ( O Q ( a 1 , ⋯ , a n ) / c ) ∗ O Q ( a 1 , ⋯ , a n ) ∗ → Cl ( Q ( a 1 , ⋯ , a n ) , c ) → Cl ( Q ( a 1 , ⋯ , a n ) ) → 0 , \begin{equation*} 0 \rightarrow \frac{(\mathcal {O}_{\mathbb {Q}(\sqrt {a_1}, \dots , \sqrt {a_n})}/c)^{*}}{\mathcal {O}_{\mathbb {Q}(\sqrt {a_1}, \dots , \sqrt {a_n})}^{*}} \rightarrow \text{Cl}(\mathbb {Q}(\sqrt {a_1}, \dots , \sqrt {a_n}),c) \rightarrow \text{Cl}(\mathbb {Q}(\sqrt {a_1}, \dots , \sqrt {a_n})) \rightarrow 0, \end{equation*} which we will call the ray class group sequence. Then one obtains from Theorem 1.1 and the ray class group sequence dim F 2 Cl ( Q ( a 1 , ⋯ , a n ) , c ) [ 2 ] ⩽ ω ( a 1 · … · a n ) · 2 n − 1 − 2 n + 1 + 2 n · ω ( c ) , \begin{equation*} \text{dim}_{\mathbb {F}_2}\text{Cl}(\mathbb {Q}(\sqrt {a_1}, \dots , \sqrt {a_n}),c)[2] \leqslant \omega (a_1 \cdot \ldots \cdot a_n) \cdot 2^{n - 1} - 2^n + 1+2^n \cdot \omega (c), \end{equation*} where the bound can be reached only if all the prime divisors of c split completely in Q ( a 1 , … , a n ) $\mathbb {Q}(\sqrt {a_1}, \ldots , \sqrt {a_n})$ . It is, once more, not obvious whether this bound is sharp. Our next theorem settles these questions. Theorem 1.4.Take n ∈ Z ⩾ 1 $n \in \mathbb {Z}_{\geqslant 1}$ and take ( k 1 , ⋯ , k n ) ∈ ( 2 · Z ⩾ 1 ) n $(k_1, \dots , k_n) \in (2 \cdot \mathbb {Z}_{\geqslant 1})^{n}$ . Let c be a squarefree positive integer divisible only by primes congruent to 1 modulo 4. Then there are infinitely many acceptable vectors ( a 1 , … , a n ) $(a_1, \ldots , a_n)$ with ω ( a i ) = k i $\omega (a_i) = k_i$ for each i ∈ [ n ] $i \in [n]$ and dim F 2 Cl ( Q ( a 1 , ⋯ , a n ) , c ) [ 2 ] = ω ( a 1 · … · a n ) · 2 n − 1 − 2 n + 1 + 2 n · ω ( c ) . \begin{equation*} \mbox{dim}_{\mathbb {F}_2}\mbox{Cl}(\mathbb {Q}(\sqrt {a_1}, \dots , \sqrt {a_n}),c)[2] = \omega (a_1 \cdot \ldots \cdot a_n) \cdot 2^{n - 1} - 2^n + 1+2^n \cdot \omega (c). \end{equation*} As a corollary of Theorem 1.4 we obtain the following result on unit groups. Corollary 1.5.Let n ∈ Z ⩾ 1 $n \in \mathbb {Z}_{\geqslant 1}$ . Let c be a squarefree positive integer with all factors congruent to 1 modulo 4. Then there exist infinitely many acceptable vectors ( a 1 , … , a n ) $(a_1, \ldots , a_n)$ such that all prime divisors of c split completely in Q ( a 1 , … , a n ) $\mathbb {Q}(\sqrt {a_1}, \ldots , \sqrt {a_n})$ and the unit group O Q ( a 1 , … , a n ) ∗ $\mathcal {O}_{\mathbb {Q}(\sqrt {a_1}, \ldots , \sqrt {a_n})}^{*}$ reduced modulo c is entirely contained in the group O Q ( a 1 , … , a n ) c O Q ( a 1 , … , a n ) ∗ 2 . \begin{equation*} {\left(\frac{\mathcal {O}_{\mathbb {Q}(\sqrt {a_1}, \ldots , \sqrt {a_n})}}{c \mathcal {O}_{\mathbb {Q}(\sqrt {a_1}, \ldots , \sqrt {a_n})}}\right)}^{*2}. \end{equation*} We remark that, in the context of Corollary 1.5, it is no real loss of generality to demand that all the prime divisors of c are 1 modulo 4. Indeed, we are aiming to construct multiquadratic extensions splitting completely at all prime divisors of c and whose unit group consists entirely of squares modulo c. This then in particular applies to −1, which is then a square in F l $\mathbb {F}_l$ for every l ∣ c $l \mid c$ so that l ≡ 1 mod 4 $l \equiv 1 \bmod 4$ . We similarly remark that the bound for the ordinary class group in Theorem 1.4 (i.e., the case c = 1 $c = 1$ ) is not sharp, whenever one of the a i $a_i$ is divisible by a prime congruent to 3 modulo 4. 2 ADDITIVE SYSTEMS For completeness we include a self-contained proof of [12, Proposition 3.1]; we claim no originality in this section. We let X 1 , ⋯ , X d $X_1, \dots , X_d$ be arbitrary non-empty finite sets and put X : = X 1 × ⋯ × X d $X := X_1 \times \dots \times X_d$ . In our application the sets X i $X_i$ will consist of acceptable integers a i $a_i$ with ω ( a i ) = k i / 2 $\omega (a_i) = k_i/2$ . A cube C is a product set Y 1 × ⋯ × Y d $Y_1 \times \dots \times Y_d$ with Y i ⊆ X i $Y_i \subseteq X_i$ and | Y i | = 2 $|Y_i| = 2$ ; in our application we can think of C as an acceptable vector ( a 1 , ⋯ , a d ) $(a_1, \dots , a_d)$ with ω ( a i ) = k i $\omega (a_i) = k_i$ . It is here that we make essential use of k i $k_i$ being even. As we see in our next section, we need to find cubes C satisfying certain bilinear conditions. The aim of our next definition is to encapsulate this in an abstract framework. We write X i 2 $X_i^2$ for the set X i × X i $X_i \times X_i$ . For S ⊆ [ d ] $S \subseteq [d]$ and i ∈ [ d ] $i \in [d]$ , π i $\pi _i$ denotes the natural projection from ∏ i ∈ S X i 2 × ∏ i ∈ [ d ] − S X i $\prod _{i \in S} X_i^2 \times \prod _{i \in [d] - S} X_i$ to X i 2 $X_i^2$ if i ∈ S $i \in S$ and to X i $X_i$ if i ∈ [ d ] − S $i \in [d] - S$ , while pr 1 $\text{pr}_1$ and pr 2 $\text{pr}_2$ denote the natural projections from X i 2 $X_i^2$ to its two factors. Definition 2.1.Let X 1 , ⋯ , X d $X_1, \dots , X_d$ be arbitrary non-empty finite sets and put X : = X 1 × ⋯ × X d $X := X_1 \times \dots \times X_d$ . An additive system A $\mathfrak {A}$ on X is a family of tuples ( C S , C S acc , F S , A S ) S ⊆ [ d ] $(C_S, C_S^{\rm{acc}}, F_S, A_S)_{S \subseteq [d]}$ , indexed by subsets S ⊆ [ d ] $S \subseteq [d]$ , satisfying the following properties: C S acc ⊆ C S ⊆ ∏ i ∈ S X i 2 × ∏ i ∈ [ d ] − S X i $C_S^{\rm{acc}} \subseteq C_S \subseteq \prod _{i \in S} X_i^2 \times \prod _{i \in [d] - S} X_i$ are sets, F S : C S → A S $F_S: C_S \rightarrow A_S$ is a map, and A S $A_S$ is a finite F 2 $\mathbb {F}_2$ -vector space; we have that C S acc : = x ∈ C S : F S ( x ) = 0 \begin{equation*} C_S^{\rm{acc}} := {\left\lbrace x \in C_S : F_S(x) = 0\right\rbrace} \end{equation*} and, for S ≠ ∅ $S \ne \emptyset$ C S : = x ∈ ∏ i ∈ S X i 2 × ∏ i ∈ [ d ] − S X i : for all j ∈ S and all y ∈ ∏ i ∈ S − { j } X i 2 × ∏ i ∈ [ d ] − ( S − { j } ) X i satisfying π k ( x ) = π k ( y ) for k ∈ [ d ] − { j } and π j ( y ) ∈ { pr 1 ( π j ( x ) ) , pr 2 ( π j ( x ) ) } , ∏ i ∈ S we have y ∈ C S − { j } acc ; \begin{align*} &C_S := \left\lbrace x \in \prod _{i \in S} X_i^2 \times \prod _{i \in [d] - S} \hspace{-7.11317pt} X_i : \mbox{ for all } j \in S \mbox{ and all } y \in \prod _{i \in S - \lbrace j\rbrace } X_i^2 \times \prod _{i \in [d] - (S - \lbrace j\rbrace )} X_i \right.\\ &\mbox{ satisfying } \pi _k(x) = \pi _k(y) \mbox{ for } k \in [d] - \lbrace j\rbrace \mbox{ and } \pi _j(y) \in \lbrace \mbox{pr}_1(\pi _j(x)), \mbox{pr}_2(\pi _j(x))\rbrace , \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \left.\vphantom{\prod _{i \in S}}\mbox{ we have } y \in C_{S - \lbrace j\rbrace }^{\rm{acc}}\right\rbrace ; \end{align*} suppose that x 1 , x 2 , x 3 ∈ C S $x_1, x_2, x_3 \in C_S$ and suppose that there exists j ∈ S $j \in S$ such that π k ( x 1 ) = π k ( x 2 ) = π k ( x 3 ) for all k ∈ [ d ] − { j } \begin{equation*} \pi _k(x_1) = \pi _k(x_2) = \pi _k(x_3) \mbox{ for all } k \in [d] - \lbrace j\rbrace \end{equation*} and π j ( x 1 ) = ( a , b ) , π j ( x 2 ) = ( b , c ) , π j ( x 3 ) = ( a , c ) for some a , b , c ∈ X j . \begin{equation*} \pi _j(x_1) = (a, b), \quad \pi _j(x_2) = (b, c), \quad \pi _j(x_3) = (a, c) \mbox{ for some } a, b, c \in X_j. \end{equation*} Then we have F S ( x 1 ) + F S ( x 2 ) = F S ( x 3 ) . \begin{align} F_S(x_1) + F_S(x_2) = F_S(x_3). \end{align} (2.1) Note that we do not quite work with cubes in the above definition, but instead with elements of X 1 × X 1 × ⋯ × X d × X d $X_1 \times X_1 \times \dots \times X_d \times X_d$ . The major difference is that we have also included elements with equal coordinates. This will be very convenient in the proof of our next counting result for C S acc $C_S^{\rm{acc}}$ . Later, we shall need to remove such elements, but it is not hard to show that they contribute a vanishingly small proportion. Proposition 2.2.Let X 1 , ⋯ , X d $X_1, \dots , X_d$ be arbitrary non-empty finite sets and put X : = X 1 × ⋯ × X d $X := X_1 \times \dots \times X_d$ . Let A $\mathfrak {A}$ be an additive system on X such that | A S | ⩽ N $|A_S| \leqslant N$ for all S ⊆ [ d ] $S \subseteq [d]$ and write δ for the density of C ∅ acc $C_\emptyset ^{\rm{acc}}$ in X. Then we have that | C [ d ] acc | ∏ i ∈ [ d ] | X i 2 | ⩾ δ 2 d · N − 3 d . \begin{equation*} \frac{|C_{[d]}^{\rm{acc}}|}{\prod _{i \in [d]} |X_i^2|} \geqslant \delta ^{2^d} \cdot N^{-3^d}. \end{equation*} Proof.We proceed by induction on d with the case d = 0 $d = 0$ being trivial (recall that the empty cartesian product is by definition the set of cardinality 1 containing the empty tuple). Fix an element x ∈ ∏ i ∈ [ d − 1 ] X i 2 . \begin{equation*} x \in \prod _{i \in [d - 1]} X_i^2. \end{equation*} Define V ( x ) : = { ( x , a ) ∈ C [ d − 1 ] acc : a ∈ X d } , W ( x ) : = { ( x , ( a , b ) ) ∈ C [ d ] acc : ( a , b ) ∈ X d 2 } . \begin{equation*} V(x) := \lbrace (x, a) \in C_{[d - 1]}^{\text{acc}} : a \in X_d\rbrace , \quad W(x) := \lbrace (x, (a, b)) \in C_{[d]}^{\text{acc}} : (a, b) \in X_d^2\rbrace. \end{equation*} By definition of an additive system, we see that W ( x ) $W(x)$ naturally injects in V ( x ) × V ( x ) $V(x) \times V(x)$ . From now on we shall identify W ( x ) $W(x)$ with its image in V ( x ) × V ( x ) $V(x) \times V(x)$ . We claim that W ( x ) $W(x)$ defines an equivalence relation on V ( x ) $V(x)$ .If we apply Equation (2.1) with a = b = c $a = b = c$ , we conclude that for all S ⊆ [ d − 1 ] $S \subseteq [d - 1]$ , all y ∈ ∏ i ∈ S X i 2 × ∏ i ∈ [ d − 1 ] − S X i $y \in \prod _{i \in S} X_i^2 \times \prod _{i \in [d - 1] - S} X_i$ and all a ∈ X d $a \in X_d$ F S ∪ { d } ( y , ( a , a ) ) = 0 . \begin{equation*} F_{S \cup \lbrace d\rbrace }(y, (a, a)) = 0. \end{equation*} From this, it follows quickly that W ( x ) $W(x)$ is reflexive. Applying Equation (2.1) with a = c $a = c$ , we then get F S ∪ { d } ( y , ( a , b ) ) + F S ∪ { d } ( y , ( b , a ) ) = F S ∪ { d } ( y , ( a , a ) ) = 0 , \begin{equation*} F_{S \cup \lbrace d\rbrace }(y, (a, b)) + F_{S \cup \lbrace d\rbrace }(y, (b, a)) = F_{S \cup \lbrace d\rbrace }(y, (a, a)) = 0, \end{equation*} so that W ( x ) $W(x)$ is symmetric. Finally, Equation (2.1) with a, b, and c arbitrary implies the transitivity of W ( x ) $W(x)$ , which establishes the claim.Our next step is to estimate the number of equivalence classes. To do so, take ( x , a ) , ( x , b ) ∈ V ( x ) $(x, a), (x, b) \in V(x)$ and { d } ⊆ S ⊆ [ d ] $\lbrace d\rbrace \subseteq S \subseteq [d]$ . Then we write ( x , a ) ∼ S ( x , b ) $(x, a) \sim _S (x, b)$ if ( x , a ) ∼ S ′ ( x , b ) $(x, a) \sim _{S^{\prime }} (x, b)$ for all { d } ⊆ S ′ ⊊ S $\lbrace d\rbrace \subseteq S^{\prime } \subsetneq S$ and F S ( y , ( a , b ) ) ) = 0 \begin{equation*} F_S(y, (a, b))) = 0 \end{equation*} for all y ∈ ∏ i ∈ S − { d } X i 2 × ∏ i ∈ [ d − 1 ] − S X i $y \in \prod _{i \in S - \lbrace d\rbrace } X_i^2 \times \prod _{i \in [d - 1] - S} X_i$ satisfying π i ( y ) = π i ( x ) $\pi _i(y) = \pi _i(x)$ for i ∈ S − { d } $i \in S - \lbrace d\rbrace$ and π i ( y ) ∈ { pr 1 ( π i ( x ) ) , pr 2 ( π i ( x ) ) } $\pi _i(y) \in \lbrace \text{pr}_1(\pi _i(x)), \text{pr}_2(\pi _i(x))\rbrace$ for i ∈ [ d − 1 ] − S $i \in [d - 1] - S$ . Note that the equivalence relation ∼ [ d ] $\sim _{[d]}$ is precisely W ( x ) $W(x)$ .To upper bound the number of equivalence classes, take a collection of points ( x , a i ) ∈ V ( x ) $(x, a_i) \in V(x)$ such that ( x , a i ) ∼ S ( x , a j ) $(x, a_i) \sim _S (x, a_j)$ for all strict subsets S of [d]. Suppose that among this collection there are R equivalence classes for ∼ [ d ] $\sim _{[d]}$ , with representatives ( x , b 1 ) , ⋯ , ( x , b R ) $(x, b_1), \dots , (x, b_R)$ . Then we see that the map ( x , b i ) ↦ F [ d ] ( x , ( b 1 , b i ) ) \begin{equation*} (x, b_i) \mapsto F_{[d]}(x, (b_1, b_i)) \end{equation*} is injective and hence we conclude that R ⩽ N $R \leqslant N$ . Therefore we conclude that the total number of equivalence classes is at most N · B , \begin{equation*} N \cdot B, \end{equation*} where B is the number of equivalence classes for the intersection of all the equivalence relations ∼ S $\sim _{S}$ , where S runs among all strict subsets of [d]. We can now iterate the above reasoning on each such S, where we have to run over all the 2 d − | S | $2^{d-|S|}$ choices of y, tracking at each step how many additional classes one obtains when passing from ∩ S ′ ⊊ S ( ∼ S ′ ) $\cap _{S^{\prime } \subsetneq S}(\sim _{S^{\prime }})$ to S. In this manner we see that the total number of equivalence classes for ∼ [ d ] $\sim _{[d]}$ is bounded by ∏ { d } ⊆ S ⊆ [ d ] N 2 d − | S | ⩽ ∏ i = 0 d − 1 N d − 1 i 2 i = N 3 d − 1 . \begin{equation*} \prod _{\lbrace d\rbrace \subseteq S \subseteq [d]} N^{2^{d - |S|}} \leqslant \prod _{i = 0}^{d - 1} N^{\binom{d - 1}{i} 2^i} = N^{3^{d - 1}}. \end{equation*} Define δ ( x ) = | V ( x ) | | X d | . \begin{equation*} \delta (x) = \frac{|V(x)|}{|X_d|}. \end{equation*} Observe that V ( x ) $V(x)$ naturally injects in X d $X_d$ by projecting on the dth component. This allows us to view V ( x ) × V ( x ) $V(x) \times V(x)$ as a subset of X d 2 $X_d^2$ . Then it follows that the density of V ( x ) × V ( x ) $V(x) \times V(x)$ in X d 2 $X_d^2$ is δ ( x ) 2 $\delta (x)^2$ . Then Cauchy's inequality and our bound for the total number of equivalence classes imply that | W ( x ) | | X d 2 | ⩾ δ ( x ) 2 N 3 d − 1 . \begin{equation*} \frac{|W(x)|}{|X_d^2|} \geqslant \frac{\delta (x)^2}{N^{3^{d - 1}}}. \end{equation*} So far we have proven that | C [ d ] acc | ∏ i ∈ [ d ] | X i 2 | = ∑ x ∈ ∏ i ∈ [ d − 1 ] X i 2 | W ( x ) | ∏ i ∈ [ d ] | X i 2 | ⩾ 1 ∏ i ∈ [ d − 1 ] | X i 2 | · ∑ x ∈ ∏ i ∈ [ d − 1 ] X i 2 δ ( x ) 2 N 3 d − 1 . \begin{align} \frac{|C_{[d]}^{\rm{acc}}|}{\prod _{i \in [d]} |X_i^2|} = \frac{\sum _{x \in \prod _{i \in [d - 1]} X_i^2} |W(x)|}{\prod _{i \in [d]} |X_i^2|} \geqslant \frac{1}{\prod _{i \in [d - 1]} |X_i^2|} \cdot \sum _{x \in \prod _{i \in [d - 1]} X_i^2} \frac{\delta (x)^2}{N^{3^{d - 1}}}. \end{align} (2.2)Another application of Cauchy's inequality shows that ∑ x ∈ ∏ i ∈ [ d − 1 ] X i 2 δ ( x ) 2 N 3 d − 1 ⩾ ∑ x ∈ ∏ i ∈ [ d − 1 ] X i 2 δ ( x ) 2 N 3 d − 1 · ∏ i ∈ [ d − 1 ] | X i 2 | . \begin{align} \sum _{x \in \prod _{i \in [d - 1]} X_i^2} \frac{\delta (x)^2}{N^{3^{d - 1}}} \geqslant \frac{{\left(\sum _{x \in \prod _{i \in [d - 1]} X_i^2} \delta (x)\right)}^2}{N^{3^{d - 1}} \cdot \prod _{i \in [d - 1]} |X_i^2|}. \end{align} (2.3)The average of δ ( x ) $\delta (x)$ over all choices of x equals the density of C [ d − 1 ] acc $C_{[d - 1]}^{\text{acc}}$ in X d × ∏ i ∈ [ d − 1 ] X i 2 $X_d \times \prod _{i \in [d - 1]} X_i^2$ , which gives the equality ∑ x ∈ ∏ i ∈ [ d − 1 ] X i 2 δ ( x ) 2 = ∏ i ∈ [ d − 1 ] | X i | 4 · ∑ a ∈ X d | { z ∈ C [ d − 1 ] acc : π d ( z ) = a } | | X d | · ∏ i ∈ [ d − 1 ] | X i 2 | 2 . \begin{align} {\left(\sum _{x \in \prod _{i \in [d - 1]} X_i^2} \delta (x)\right)}^2 = \prod _{i \in [d - 1]} |X_i|^4 \cdot {\left(\frac{\sum _{a \in X_d} |\lbrace z \in C_{[d - 1]}^{\text{acc}} : \pi _d(z) = a\rbrace |}{|X_d| \cdot \prod _{i \in [d - 1]} |X_i^2|}\right)}^2. \end{align} (2.4)For each a ∈ X d $a \in X_d$ , consider the additive system A a $\mathfrak {A}_a$ on X 1 × ⋯ × X d − 1 $X_1 \times \dots \times X_{d - 1}$ obtained from the additive system A $\mathfrak {A}$ in the statement of the proposition and a in the natural way. The induction hypothesis applied to each A a $\mathfrak {A}_a$ yields | { z ∈ C [ d − 1 ] acc : π d ( z ) = a } | ∏ i ∈ [ d − 1 ] | X i 2 | ⩾ δ a 2 d − 1 · N − 3 d − 1 , \begin{align} \frac{|\lbrace z \in C_{[d - 1]}^{\text{acc}} : \pi _d(z) = a\rbrace |}{\prod _{i \in [d - 1]} |X_i^2|} \geqslant \delta _a^{2^{d - 1}} \cdot N^{-3^{d - 1}}, \end{align} (2.5)where δ a $\delta _a$ is the density of { π [ d − 1 ] ( z ′ ) : z ′ ∈ C ∅ acc , π d ( z ′ ) = a } $\lbrace \pi _{[d - 1]}(z^{\prime }) : z^{\prime } \in C_\emptyset ^{\text{acc}}, \pi _d(z^{\prime }) = a\rbrace$ in ∏ i ∈ [ d − 1 ] X i $\prod _{i \in [d - 1]} X_i$ . Since d ⩾ 1 $d \geqslant 1$ , the generalized mean inequality (with 2 d − 1 ⩾ 1 $2^{d - 1} \geqslant 1$ ) shows that ∑ a ∈ X d δ a 2 d − 1 | X d | 2 ⩾ ∑ a ∈ X d δ a | X d | 2 d = δ 2 d . \begin{align} {\left(\frac{\sum _{a \in X_d} \delta _a^{2^{d - 1}}}{|X_d|}\right)}^2 \geqslant {\left(\frac{\sum _{a \in X_d} \delta _a}{|X_d|}\right)}^{2^d} = \delta ^{2^d}. \end{align} (2.6)We deduce from Equations (2.2)–(2.6) that | C [ d ] acc | ∏ i ∈ [ d ] | X i 2 | ⩾ δ 2 d N 3 d − 1 · N − 2 · 3 d − 1 = δ 2 d · N − 3 d \begin{equation*} \frac{|C_{[d]}^{\rm{acc}}|}{\prod _{i \in [d]} |X_i^2|} \geqslant \frac{\delta ^{2^d}}{N^{3^{d - 1}}} \cdot N^{-2 \cdot 3^{d - 1}} = \delta ^{2^d} \cdot N^{-3^d} \end{equation*} as desired. □ $\Box$ 3 THE CATEGORY OF EXPANSION GROUPS In this section we summarize the main results from [7] that we will use in Section 4. We start by introducing n-expansion groups. For the motivation behind our next definition and Definition 3.3, see Proposition 4.1. Definition 3.1.Write