Density of type-dependent sets in Krull monoids with analytic structure

We describe structural and quantitative properties of type-dependent sets in monoids with suitable analytic structure, including simple analytic monoids, introduced by Kaczorowski (Semigroup Forum 94:532–555, 2017. https://doi.org/10.1007/s00233-016-9778-9), and formations, as defined by Geroldinger and Halter-Koch (Non-unique factorizations, Chapman and Hall, Boca Raton, 2006. https://doi.org/10.1201/9781420003208). We propose the notions of rank and degree to measure the size of a type-dependent set in structural terms. We also consider various notions of regularity of type-dependent sets, related to the analytic properties of their zeta functions, and obtain results on the counting functions of these sets.


Introduction
The goal of quantitative factorization theory is to describe, as precisely as possible, the distribution of elements of a monoid subset A defined by factorization-related conditions. The monoid must be equipped with a norm function · making it possible to define the counting function In the present paper we study the growth of A(x) for a class of reduced monoids with divisor theory S ⊆ F(P), called shifted formations, defined in Sect. 2 of an element a in A ⊆ F(P) depends on the type of factorization of a in F(P). The kind of result that we expect is that A(x) follows an asymptotic law of the type where the numbers b, c and d depend on the structural properties of A. The goal of this paper is to introduce appropriate functions describing the structure of A, to determine their properties, and to show that versions of (2) hold for certain classes of subsets of the divisor monoid, that we call regular, almost-regular, semi-regular, etc., cf. Theorem 35. We attempt to provide an easily extendible general framework to show regularity for a variety of sets.
As examples of applications we show: a technical, but still flexible result (Proposition 37) about the regularity of sets of a specific form, and the asymptotics for the counting functions of four specific sets, given below. The result on F k is classical, although it was never considered in this particular setting.
Theorem 1 Let S be a shifted formation with a principal shift λ ≥ 0, S the set generated by absolutely irreducible elements of S (i.e. irreducibles which are powers of prime divisors [3]), S 1 the set of products of distinct absolutely irreducible elements of S, and E k the set of k-powerful divisors, where k ∈ N. We have would make the quantitative theory more complex. The author hopes to address this problem in a future paper. Quantitative factorization theory was initiated by Fogels [1] and further developed by Narkiewicz and other authors, cf. Geroldinger and Halter-Koch [2, Chapter 9] and the references cited there. Narkiewicz [6] was the first to treat type-dependent sets, including F k , that do not necessarily belong to the smaller class of -sets (see the end of Sect. 3 for a definition of -sets). He introduced the notion of depth and proposed an inductive reasoning to find the asymptotics of the counting function of a type-dependent set of height 1 whose elements have bounded depth. In the language of the present paper the sets treated in [6], subsets of rings of integers in algebraic number fields, are algebraic products of: the set of elements with all prime divisors in the principal class, and an arbitrary set of height 1 and rank 0 whose elements have no principal divisors. The idea to use induction over depth is also used in the present paper (in Lemma 22). Unfortunately, Lemma 2 in [6] only applies to a set of elements of a single type. In the proof of Corollary 4 in [6], where sets with an infinite number of types may arise, the author only mentions that they can be dealt with in the same way. Kaczorowski [4,Theorem 3] obtained the asymptotics for the counting function of S (another "properly" type-dependent set), in the context of algebraic number fields, with C explicitly determined, and an upper bound for the error term.
Geroldinger and Halter-Koch [2, Theorem 9.1.2] gave a more general result on type-dependent sets. They introduced the height, i.e. the most important of the metrics of type-dependent sets, although not explicitly as set metrics, cf. Definition 9.1.1 and the formula for e 0 in Theorem 9.1.2. They developed a number of ideas from [6], adapting them to the setting of (quasi-)formations and sets of height greater than 1. Sets treated in [2, Theorem 9.1.2] are algebraic products of three type-dependent components (with elements in distinct components relatively prime) that we describe as follows: a set C of height e 0 , a set B 1 of all elements with all prime divisors in a subset U 1 ⊆ Cl(S) and exponents divisible by e 1 , and a set B 2 of all elements with all prime divisors in a subset U 2 ⊆ Cl(S) and exponents greater or equal to e 2 . Moreover e 2 > min(e 0 , e 1 ) and rk(C) = 0 unless e 0 > e 1 . Bearing in mind the different setting, Theorem 9.1.2 can be applied directly to the set F k of Theorem 2, but not to S, S 1 or E k of Theorem 1. Asymptotic lower and upper bounds for S(x) of the correct order are given instead, cf. [2, p. 633]. The set E k is of the same general shape as required in Theorem 9.1.2, with B 2 = E k and C = B 1 = {1}, however, it does not meet the technical assumption e 2 > min(e 0 , e 1 ). In their proof of [2, Theorem 9.1.2] the authors attempted to close the gap left in [6]. Unfortunately, their argument also contains gaps, namely, on page 619, line 3, they do not take into account the repetitions that may occur when multiplying two Dirichlet series, and a similar problem occurs in the fourth display from bottom on the same page. For comparison, our Proposition 37 implies Theorems 1 and 2. It does not quite re-prove [2, Theorem 9.1.2], because we do not determine the constant factors in the asymptotic relationship ∼.
The paper is organized as follows. Section 2 contains known definitions and notation. Section 3 provides the language to describe the structural "size" of a typedependent set. Notions of rank and degree, previously defined for -sets, are extended to type-dependent sets. We show the basic properties of rank, degree and height, and how they behave under set operations like disjoint union and algebraic product. In Sect. 4 we define regular type-dependent sets, whose zeta functions have appropriate analytic properties, agreeing with the set's height, rank and degree. We also consider other, weaker and stronger regularity properties, and show how they behave under basic set operations. In Sect. 5 we show that type-dependent sets of rank 0 are regular (Theorem 23). In Sect. 6 we construct a large family of regular sets (Theorem 30).
In Sect. 7 we apply the results of previous sections to obtain asymptotics for various counting functions. Theorem 35 gives asymptotics for the counting functions of almost regular sets and bounds for semi-regular sets. In the same theorem we also give upper and lower bounds for A(x) for general type-dependent sets, using Theorems 30 and 23 respectively. It follows from Theorem 36 that these bounds cannot be improved in general, at least not using the language of the present paper. We also prove Proposition 37 and Theorems 1 and 2.

Preliminaries
We denote by N, N 0 , Z, R and C respectively the sets of positive integers, non-negative integers, integers, real numbers and complex numbers, and by s = σ + it a complex variable with real part σ and imaginary part t. We make use of Landau's O and o, and Vinogradov's notation. We write f g for f g and g f . We write f ∼ g for (1)), x → +∞. Function support is denoted as Supp, symmetric difference of sets as , the cardinality of A as |A| or # A. The infimum of the empty set is assumed to equal +∞. When G is a finite abelian group we let E denote the neutral element, ord(X ) the order of X ∈ G, and G the group of characters of G, with χ 0 for the trivial character. For U ⊆ G the subgroup generated by U is denoted as U , while χ | U = |G| −1 X ∈U χ(X ) for χ ∈ G. We compare functions α : G → N 0 using the product order, so β ≤ α means β : G → N 0 and β(X ) ≤ α(X ) for all X ∈ G. For comparing pairs and triples we use lexicographic order with the first term always being the most significant.
Suppose λ ≥ 0 and S is a Krull monoid contained in a free semigroup F(P) generated by a set of primes P such that: (ii) every p ∈ P gcd(a 1 , . . . , a n ) for some a 1 , . . . , a n ∈ S.
Hence the inclusion map S ⊆ F(P) is a divisor theory for S. Let G be the quotient of the groups generated by F(P) and S. The intersection of an element of G with F(P) is called a divisor class. We let Cl(S) denote the divisor class group, i.e. the set of divisor classes with multiplication induced by that in G. We let h denote the number of divisor classes and E the principal class. For χ ∈ Cl(S) and a ∈ F(P) we write χ(a) instead of χ([a]). We assume: (iii) the order h of the class group Cl(S) is finite.
In particular, for every χ ∈ Cl(S) the Dirichlet series For f : A → C such that for every ε > 0 we have f (a) a ε on A, we also put We make use of auxiliary functions For e ∈ N we let A e denote the set of Dirichlet series absolutely convergent for σ > (1 + λ)/e. We also use the classical function ω(a) = p∈P p|a 1. The following lemma shows, in particular, that P X / ∈ A 2 , so the set P ∩ X is infinite for every X ∈ Cl(S). Its countability follows from condition (vi).
Lemma 3 Let X ∈ Cl(S). We have Proof We have The assertion follows from Factorization-related properties of an element a ∈ S, such as uniqueness of factorization to irreducibles, factorization lengths, etc., depend, in general, on its factorization in F(P). Each element a ∈ F(P) has a unique factorization where almost all the v p (a) vanish. We say that elements a, b ∈ F(P) have the same factorization type, and write a ≈ b, if for each X ∈ Cl(S) there is a permutation π of for all p ∈ P ∩ X . We can think of "factorization type" as an equivalence class of the relation ≈. This definition is equivalent to that of a normalized type given by Geroldinger and Halter-Koch [2, Definition 3.5.7], cf. also Narkiewicz [6, (1) and 3.(a)]. We call a subset A ⊆ F(P) type-dependent if it is closed upon ≈. We say that A is trivial if A = ∅ or A = {1}.

Metrics of a type-dependent set
For a ∈ F(P) we call the (minimal) height of a. This is similar to [2, Definition 9.1.1], except γ (1) = +∞. For A ⊆ F(P) we define the (minimal) height as The height of A is therefore finite if and only if A is non-trivial. Height should be thought of in terms of sparsity, not size of a set. Indeed, the height of a subset is always less or equal to the height of a superset. We define a related measure and call it the exponential density of A. Following [2, Definition 9.1.1] we put, for a ∈ F(P), e ∈ N, X ∈ Cl(S), Narkiewicz [6] It follows from the infinitude of primes in each class that γ ( e (U , α)) = e whenever U = ∅ or α = 0. If α = 0, we have γ (a) = e for all a ∈ e (U , α). If α = 0, then e (U , α) contains all elements of height ≥ e + 1. Let A ⊆ F(P) be a non-trivial, type-dependent set of height e. We define the rank of A, denoted rk(A), as the smallest r such that A is contained in a finite union where |U i | ≤ r for i = 1, . . . , m. We also define the degree of A, denoted deg(A), as the supremum of all d ∈ N 0 such that the intersection e (U , α) ∩ A is of the same height and rank as A for some U and α such that X α(X ) = d. For trivial sets we put: In [7] the present author defined rank and degree for a subclass of type-dependent sets called -sets. We show at the end of the section that the notions defined here extend Proof Follows from the definition of height, rank and degree.

Lemma 5 The intersection of two cubes
is either empty or is itself a cube. If e > e and α = 0, then A = ∅. If e > e and

then A is non-empty if and only if
Proof Follows from the definition of a cube.

Fact 6 If A ⊆ F(P) is a non-trivial type-dependent set of height e and rank r , then deg(A) equals the smallest d such that A is contained in a finite union (4) with |U i | ≤ r for all i and X
, d be as in the assertion, and suppose A is contained in (4) with |U i | ≤ r for all i and d = max i:|U i |=r X / ∈U i α i (X ). We may assume that the choice of pairs (U i , α i ) is minimal, i.e. the number of i with |U i | = r is the smallest possible, and each intersection A ∩ e (U i , α i ) is non-empty. Moreover we can suppose that contrary to the minimality of the choice of pairs Since rk ( e (U , α) ∩ A) = r we must have (5) for at least one i, impying that . It follows from Fact 6 that every non-trivial type-dependent set has a fair covering and that in fact max i: The rank and degree cannot both be equal to zero.
Proof Let e = γ (A). The assertion follows from the existence of a fair covering (6). If U i = ∅ and α i = 0 for all i, then γ (a) > e for all a ∈ A, contrary to the assumption.
Otherwise γ (A) = e and rk(A) ≤ r . Moreover there exists a fair covering (6). The covering Proof Let e = γ (A). We may suppose that A is non-trivial. The equality γ (B) = me follows from the definition. The other equalities follow from the fact that a ∈ e (U , α) is equivalent to a m ∈ me (U , α), so (6)

is a fair covering of A if and only if
is a fair covering of B. Proof The assertion is easy to check if A is a trivial set, so we assume that A is nontrivial, hence so is B. The equality γ (AB) = γ (B) follows by taking any b ∈ B with γ (b) = γ (B), and any a ∈ A relatively prime to b (again, it exists by the infinitude of primes in each class and the fact that A is type-dependent). Then

Proposition 10 Let
, p | a for some a ∈ A, and q | b for some b ∈ B, then there exists a ∈ F(P), a ≈ a, such that q | a. Since A is type-dependent, we have a ∈ A and q | gcd(a , b), contrary to the assumptions. Therefore the sets U and V , of classes of prime divisors of elements of A and B, respectively, are disjoint. Let e = γ (B). Consider a fair covering If γ (A) = e, let (6) be a fair covering, otherwise let m = 1, U 1 = ∅, and α 1 = 0. In any case we have Our goal is to show that γ (AB) = e, rk(AB) = r +rk(B), and deg(AB) = d+deg(B). We may assume that This implies (rk(AB), deg(AB)) ≤ (r + rk(B), d + deg(B)) in the lexicographic order, where the first coordinate is more significant. It also follows from (7) that if AB is covered with sets of the form e (W , η), we can replace each e (W , η) with the union of skipping the empty summands. If non-empty, the intersection (8) where, for all k, in the lexicographic order. Let I and J be the sets of those k, for which the first, respectively the second, inequality in (10) is sharp. By Fact 6 (and by . The covering (9) is fair, so this implies the converse inequality (rk(AB), deg(AB)) ≥ (r + rk(B), d + deg(B)).
Next we show that when A is a special kind of type-dependent set called -set, as defined in [7], the values of rank and degree introduced there agree with the ones defined here. We do that without re-introducing the language of [7], only a bare minimum. For a ∈ F(P) and X ∈ Cl(S) let

Proposition 11 Let A be a non-trivial -set. Then γ (A) = 1. The rank of A equals the smallest r such that A is contained in a finite union of the form
where X i ∈ Cl(S) and |U i | ≤ r for i = 1, . . . , n. The degree of A equals the supremum d of all d ∈ N 0 such that Proof Let r = rk(A) and d = deg(A). Let r and d be as in the assertion and consider the smallest d such that A is contained in a finite union (12) , then, by the infinitude of primes in each class, there exists b ∈ F(P) which is a product of distinct primes and satisfies (11 Hence r ≤ r and, by Fact 6, also (r , d) ≤ (r , d ) in the lexicographic order. On the other hand, if A is contained in (4) with e = 1, and a ∈ A, we can find (again) b ∈ A which is a product of distinct primes and satisfies (11). Let i be such that This implies (r , d ) ≤ (r , d) in the lexicographic order. The equality d = d follows from [7,Lemma 4.4] and the basic properties of the "old" rank and degree proved there.
where e = γ (A), H i ∈ A e+1 , w i,χ ∈ C, and k i,χ ∈ N 0 for all i, χ, moreover the limit where σ 0 = ed(A), is finite and non-zero. In addition the trivial sets are also considered regular. We call A regular across classes if for every X ∈ Cl(S) such that X ∩ A = ∅ the intersection X ∩ A is regular with metrics(X We have

Lemma 13 Suppose A is a type-dependent set and
for some σ 0 = (1 + λ)/e, e ∈ N, and r , d ∈ N 0 not both zero. If γ (A) > e or A is regular, then Proof The assertion is obvious when A = ∅, so we assume that A = ∅. If γ (A) > e, then lim s→σ + 0 Z A (s) is finite by Lemma 12. This implies the assertion. Suppose γ (A) = e. Then the limit (14) is finite and either r > rk(A) or r = rk(A) and d > deg(A). Therefore so the assertion holds again.

Fact 14 Suppose A is a type-dependent set. If A is regular (respectively completely regular across classes), then so is A B for every type-dependent set B satisfying γ (B) > γ (A).
Proof Suppose A is regular and B = ∅. We have metrics(A B) = metrics(A) by Fact 8. Both B \ A and A ∩ B are of greater height than A by Fact 4. Lemma 12 and imply that Z A B (s) is of the form (13). Fact 7 and Lemma 13 show that the limit (14) for Z A B (s) is the same as for Z A (s). By replacing A and B with X ∩ A and X ∩ B and using Fact 4 we obtain the assertion for sets completely regular across classes.

Fact 17 Suppose A is a type-dependent set. If A is completely regular across classes, it is regular across classes. If it is regular across classes, it is regular. If it is regular, it is almost regular. If it is almost regular, it is semi-regular.
Proof By Fact 16 if X ∩ A is regular for every X ∈ Cl(S), then so is the disjoint union This implies the second assertion. The others are obvious.

Proof Suppose
Hence B is regular. Let Y ∈ Cl(S) and let X 1 , . . . , X n be all the solutions of X m = Y . If A is regular across classes, then n j=1 X j ∩ A is regular by Fact 16 and so the second part of the assertion follows from the first one.

Proposition 20 Let A, B ⊆ F(P) be type-dependent sets such that gcd(a, b) = 1 for all a ∈ A, b ∈ B. If A and B are regular (respectively regular across classes, completely regular across classes), then so is AB. If γ (B) > γ (A) and A is regular (respectively regular across classes, completely regular across classes), then so is AB.
Proof = lim

Z AB (s) = Z A (s)Z B (s), σ > 1 + λ,
and Z B (s) ∈ A e+1 , so Z AB (s) has the required form. We also have is either empty or regular with metrics equal to metrics(AB). If, in addition, A is completely regular across classes and B is non-empty, we have Y ∩ B = ∅ for at least one Y = Y 0 , so the right-hand side of (15) always has a non-empty summand for X = ZY −1 0 . Therefore AB is completely regular across classes.

Case 4. A is regular across classes and γ (B) > e.
This is analogous to Case 3.

Sets of rank 0
and let f : and Z A (s, f ) is a polynomial in (P X (es)) X ∈Cl(S) with coefficients in A e+1 , of degree ≤ d, and no other terms of degree d than where To see that (21) Finally, note that for each X ∈ Cl(S), δ = δ e,X (b), we have  Z A(m 1 ,...,m r ) (s) is a polynomial of degree k ≤ δ − 1 < d. Otherwise we have r = δ and m 1 = . . . = m δ = 1. Comparing the products of both sides of (22) over all X ∈ Cl(S) we conclude that is a polynomial of degree ≤ d − 1. This, in view of (19), implies our assertion.
We conclude that By Lemma 22 with f = 1, and Lemma 3, the function Z A i (s) is a polynomial of degree ≤ d in log L(es, S, χ), χ ∈ Cl(S). Hence where H i ∈ A e+1 and k j,χ ∈ N 0 , as in (13), moreover χ k j,χ ≤ d for all j. It also follows from Lemmas 22 and 3 that the sum of terms of (24) with k j,χ 0 = d equals where C = δ i =d C i . Each function log L(es, S, χ), χ = χ 0 , is regular at σ 0 = ed(A). Therefore the limit (14) equals h −d Z C (σ 0 ) > 0.

Sets generated by cubes
Lemma 24 Let e, l ∈ N, X ∈ Cl(S), Otherwise Let G denote the set of those χ that satisfy (27) and σ 0 = (1 + λ)/e. We have Z B (s, χ) = Z B (s) and χ(Y ) = 1 for χ ∈ G, so Moreover The factor exp R χ 0 (s) is regular and non-zero at σ 0 and so is  (26). If m = gcd(e, l) it is sufficient to prove the regularity of A e/m,l/m,X across classes and apply Fact 19. Therefore we assume that gcd(e, l) = 1. We have, for every χ ∈ Cl(S) and σ > 1 + λ, where the last equality follows from the power series identity Hence where C = a ∈ F(P) : p ∈ X and v p (a) ≥ e + l for every p | a, p ∈ P and f : C → R is a multiplicative function defined by for p ∈ X ∩ P. We have γ (C) = e + l and f ( p j ) < j ≤ log 2 p j , hence Z C (s, χ f ) ∈ A e+1 by Lemma 12. Let Y ∈ Cl(S) be such that Y ∩ A = ∅. We have Y ∈ X . Similarly to the proof of Lemma 25 we have where R χ (s) ∈ A e+1 and χ e | {X } < 1/h unless χ satisfies (27). It follows that Z Y ∩A (s) is of the required form. Again, let G denote the set of χ satisfying (27). We have is finite and real. By Lemma 25 the limit lim s→σ + The summand with r = 0 is either 1 or 0, depending on whether Y ∈ X e or not. For r ≥ 1, a ∈ C and ω(a) = r we have a = p k 1 1 . . . p k r r for some k 1 , . . . , k r ∈ N and distinct p 1 , . . . , p r ∈ X ∩ P. Moreover, a ∈ Y is equivalent to k 1 + . . . + k r ≡ k (mod e), where k is such that Y = X k . Therefore  (U , α )).

Theorem 30 Let A be the algebra of subsets of F(P) generated by all cubes with the binary operations of set union, intersection and difference. Every non-empty set A ∈ A is completely regular across classes.
Proof First we show by induction that every set of the form is completely regular across classes. Let D be given by (29). For m = 0 the assertion follows from Lemma 28. Suppose m ≥ 1 and the assertion holds for m − 1. Suppose D = ∅. Note that so, by Lemma 5 and D = ∅ we may assume α) and, by Corollary 29, metrics( e i (U i , α i )) < metrics( e (U , α)) for each i. It follows from Fact 8 that metrics(D) = metrics( e (U , α)). Let We have D = A \ B. By the inductive hypothesis A is completely regular across classes, and so is B, unless B = ∅. In the latter case we are done, so suppose B = ∅.

Corollary 31 Every non-trivial type-dependent set A is contained in a type-dependent set B, completely regular across classes, with metrics(B) = metrics(A).
Proof Let (6) be a fair covering of A and let B be the right-hand side of (6). We have metrics(B) = metrics(A) and the regularity of B follows from Theorem 30.

Corollary 33
The assumption "if γ (A) > e or A is regular" in Lemma 13 may be dropped.
Proof It is enough to apply Lemma 13 to a regular superset A of A such that metrics(A ) = metrics(A). The assertion for A follows from |Z A (σ )| ≤ |Z A (σ )|.

Corollary 34 If A is an almost regular set satisfying (13), then A is regular.
Proof Let B ⊆ F(P) be such that metrics (B) < metrics(A) and A B is regular. The assertion follows from: and Corollary 33.

Counting functions of regular sets
Theorem 35 Let A be a non-trivial type-dependent set. If A is almost regular, then and for every M > 0.
Proof When A is regular with σ 0 = ed(A), it follows from (13) that we have We may assume that m is the smallest possible, hence the pairs and the trigonometric polynomial m i=1 H i (σ 0 )(σ − σ 0 ) −i w i is not identically zero, the exponents −i w i being all distinct. By (14) we have w = rk(A)/h, k = deg(A), m = 1 and w 1 = w. The first assertion now follows from the Tauberian theorem of Delange and Ikehara, e.g., in the form given in [2, Theorem 8.2.5] that we can apply to Z A (σ 0 s), because σ 0 > 0. Suppose A is almost regular and A B is regular for some B ⊆ F(P) such that metrics (B) < metrics(A), then let B be a regular superset of B satisfying metrics(B ) = metrics(B), that exists by Corollary 31. We have metrics(A B) = metrics(A) by Fact 8, so for some C > 0 and Since Proof We construct an ascending sequence of type-dependent sets (A n ) of height e and rank 0 and an ascending sequence of positive numbers (x n ). Let A 1 = { p e : p ∈ P}. When A n is constructed we conclude from Theorems 23 and 35 that A n (x) = o( f (x)), x → +∞, so we can find x n > n such that A n (x) ≤ f (x)/n for all x ≥ x n . Let m = m n = log 2 (x n ) and A n+1 = A n ∪ p e 1 . . . p e m : p 1 , . . . , p m ∈ P We have a ≥ 2 m for all a ∈ A n+1 \ A n , so A n+1 (x) = A n (x) for all x ≤ x n . The fact that A n+1 is also of rank 0 follows from Facts 8 and 21. Then A = ∞ n=1 A n satisfies and lim n→∞ x n + ∞. It remains to show that rk(A) = h. Suppose the contrary and let (6) be a fair covering with max i |U i | < h. Let M = max i X α i (X ) and let n be such that x n > 2 h(M+1) . Then m n > h(M + 1) and we can find a ∈ A n+1 such that δ e,X (a) > M + 1 for every X ∈ Cl(S). Let i be such that a ∈ e (U i , α i ) and let X be such that X / ∈ U i . Then where we understand inf X ∈U α(X ) as +∞ when U = ∅. If γ (C) > inf X ∈U α(X ) or metrics(X ∩ C) = metrics(C) for every X ∈ Cl(S) such that X ∩ C = ∅, then A is regular across classes. If γ (C) > γ (B), the assertions follow from Propositions 20 and 10. Otherwise rk(C) = 0, so C is regular by Theorem 23 and so is Y ∩ C for every Y ∈ Cl(S) such that γ (Y ∩ C) = γ (C). If metrics(X ∩ C) = metrics(C) whenever X ∩ C = ∅, then C is regular across classes. By Proposition 20 the set A is regular, and if C is regular across classes, then A is regular across classes. The assertions about metrics(A) follow from Lemma 21 and Proposition 10. Suppose h > 1. Then for every X ∈ Cl(S) \ {E} and p 1 , . . . , p ord(X ) ∈ X ∩ P we have p 1 . . . p ord(X ) ∈ C, so γ (C) = 1. We note, following Narkiewicz, that if p 1 , . . . , p (k+1) ord(X ) ∈ X ∩P are distinct primes dividing a ∈ S and X ∈ Cl(S)\{E}, then p 1 , . . . , p (k+1) ord(X ) can be divided to k + 1 groups of ord(X ) elements in ((k + 1) ord(X ))! (ord(X )!) k+1 > k ways, giving rise to more than k distinct factorizations of a. Hence for every a ∈ C we have X δ 1,X (a) ≤ ω(a) < h(h − 1)(k + 1), so rk(C) = 0 and N k := deg(C) < h(h − 1)(k + 1). Otherwise, if h = 1, we have C = {1} and F k = B, and we put N k := deg(C) = 0. In either case the assumptions of Proposition 37 are therefore satisfied with U = {E} and α(E) = β(E) = 1. The assertion follows from Theorem 35.