New York Journal of Mathematics

New York J. Math. 8 (2002) 63–83.

Asymptotic Density in Combined Number Systems

Karen Yeats

Abstract. Necessary and sufficient conditions are found for a combination of additive number systems and a combination of multiplicative number systems to preserve the property that all partition sets have asymptotic density. These results cover and extend several special cases mentioned in the literature and give partial solutions to two problems in [6].

Contents

1. Introduction 63 2. Preliminaries 64 3. The nice cases 66

3.1. Additive 66 3.2. Multiplicative 70

4. The general situation 74 4.1. Additive 74 4.2. Multiplicative 78

References 82

1. Introduction

The recent ancestry of the problems examined in this paper began in 1937 when abstract number systems with real valued norms were introduced by Arne Beurling [5] for the purpose of finding minimal sufficient conditions to prove prime number theorems. This is also the subject of Paul Bateman and Harold Diamond’s article [2]. John Knopfmacher focused on density questions for classes of well known algebraic and topological structures in his books [12] and [13].

Received June 27, 2001. Mathematics Subject Classification. Primary 11N45, 11P99, Secondary 05A16, 11N80. Key words and phrases. additive number system, multiplicative number system, asymptotic

density, partition set, ratio test, regular variation.

Thanks to Stan Burris, a patient and meticulous supervisor, to the referee for his interest and detailed comments, and to NSERC for their Undergraduate Student Research Award which supported this research.

63^{ISSN 1076-9803/02}

Interest in number systems with the property that is central to the discussion below, namely that all partition sets have asymptotic density, started with Kevin Compton’s work relating combinatorics to logic [9], [10], [11]. He proves logical limit laws on certain classes of structures using enumeration methods. Stanley Burris’s book Number Theoretic Density and Logical Limit Laws [6] pulls the abstract number systems and the logical asymptotic combinatorics together.

In [6] Burris poses the following problem (Problem 5.20): Find necessary and sufficient conditions on two additive number systems A_{1} and A_{2} for all partition sets of A_{1}∗A_{2} to have asymptotic density. He also poses the corresponding problem for multiplicative number systems (Problem 11.25): Find necessary and sufficient conditions on two multiplicative number systems A_{1} and A_{2} for all partition sets of A_{1} ∗ A_{2} to have global asymptotic density.^{1}

This paper considers these problems when A_{1} and A_{2} themselves have the property that all partition sets have asymptotic density. Elegant and popular special cases for additive and multiplicative systems will be considered first, followed by the general cases.

2. Preliminaries

Definition 1. ℕ = {0, 1, 2, … }.

Definition 2. A number system

A = (A, P, ∗, e, ∥ ∥)

consists of a countable free commutative monoid (A, ∗, e), where P is the nonempty set of indecomposable elements, and ∥ ∥ a norm.

Definition 3 ([6], 2.5 and 2.7). An additive number system is a number system for which ∥ ∥ is an additive norm, that is, a mapping from A to the nonnegative integers such that ∥a∥ = 0 iff a = e, ∥a ∗ b∥ = ∥a∥ + ∥b∥, and for every n ∈ ℕ the set {a ∈ A : ∥a∥ = n} is finite.

Definition 4 ([6], 8.1 and 8.2). A multiplicative number system is a number system for which ∥ ∥ is a multiplicative norm, that is, a mapping from A to the positive integers such that ∥a∥ = 1 iff a = e, ∥a∗b∥ = ∥a∥·∥b∥, and for every positive integer n the set {a ∈ A : ∥a∥ = n} is finite.

Definition 5 ([6], 2.12 and 2.28). Given a number system A, for each set B ⊆ A the (local) counting function of B is

b(n) = |{b ∈ B : ∥b∥ = n}|,

and the global counting function of B is

B(x) = ∑_{n≤x} b(n).

Notice that B(x) is nondecreasing. As special cases we have a(n) and p(n), the (local) counting functions of A and P, respectively, and A(x), the global counting function of A. We will refer to a(n) and p(n) as the (local) counting functions of A and to A(x) as the global counting function of A.

1Burris in [6] uses + for the combination of additive number systems and × for the combination of multiplicative number systems.

Definition 6 ([6], 3.21, 9.20). For A a number system and B_{1}, …, B_{k} ⊆ A, let

B_{1} ∗ ⋯ ∗ B_{k} = {b_{1} ∗ ⋯ ∗ b_{k} : b_{i} ∈ B_{i}}.

Definition 7 ([6], 3.23, 9.22). For A a number system and B ⊆ A,

B^{0} = {e}
B^{m} = \underbrace{B ∗ ⋯ ∗ B}_{m times} for m > 0
B^{≤m} = B^{0} ∪ ⋯ ∪ B^{m} for m ≥ 0
B^{≥m} = ⋃_{n≥m} B^{n} for m ≥ 0.

Definition 8 ([6], 3.25, 9.24). Given a number system A a subset B of A is a partition set of A if B can be written in the form

B = P_{1}^{γ_{1}} ∗ ⋯ ∗ P_{k}^{γ_{k}}

where P_{1}, …, P_{k} is a finite partition of the set P of indecomposables of A, and each γ_{i} is of the form m, ≤ m, or ≥ m.

Definition 9 ([6], 4.14, 10.4). Given number systems A_{1} and A_{2}, both additive or both multiplicative, define A_{1} ∗ A_{2} to be the number system whose underlying monoid is the direct product of the two monoids, and if A_{1} and A_{2} are additive then the norm is the sum of the coordinate norms, that is, ∥(a_{1}, a_{2})∥ = ∥a_{1}∥_{1} + ∥a_{2}∥_{2}, while if A_{1} and A_{2} are multiplicative then the norm is the product of the coordinate norms, that is, ∥(a_{1}, a_{2})∥ = ∥a_{1}∥_{1} · ∥a_{2}∥_{2}.

Note that A_{1} ∗ A_{2} is additive if A_{1} and A_{2} are additive and is multiplicative if A_{1} and A_{2} are multiplicative. The set of indecomposables of A_{1} ∗ A_{2} is

{(p_{1}, e) : p_{1} ∈ P_{1}} ∪ {(e, p_{2}) : p_{2} ∈ P_{2}}.

Let a_{i}(n) and p_{i}(n) be the local counting functions of A_{i} for i = 1, 2, and let a(n) and p(n) be the local counting functions of A_{1} ∗ A_{2}. Then

p(n) = p_{1}(n) + p_{2}(n)

a(n) = \begin{cases}∑_{i+j=n} a_{1}(i)a_{2}(j) & for A an additive number system \\ ∑_{i·j=n} a_{1}(i)a_{2}(j) & for A a multiplicative number system. \end{cases}

Lemma 10. Let A = A_{1}∗A_{2} be a multiplicative number system and let B = B_{1}×B_{2} where B_{i} ⊆ A_{i}. Then

B(x) = ∑_{1≤k≤x} b_{1}(k)B_{2}(x/k).

Proof. Theorem 3.10 from [1]. □

Lemma 11. Let A_{1} and A_{2} be number systems both additive or both multiplicative. Then any partition set B of A_{1} ∗ A_{2} can be written in the form

B = ⋃_{j=1}^{k} B_{1j} × B_{2j}

for some k ≥ 1 where the union is disjoint and each B_{ij} is a partition set of A_{i}.

Proof. The proof is routine and thus is left to the reader to complete by using the definitions of m, ≥ m, and ≤ m, and the fact that if B, C, and D are sets of numbers from a number system, then (B ∪ C) ∗ D = (B ∗ D) ∪ (C ∗ D). □

3. The nice cases

In this section we will deal with the most popular additive and multiplicative systems, namely the reduced additive systems and the strictly multiplicative systems. In Section 4 we look at the general situation.

3.1. Additive. Throughout the additive subsections of this paper A, A_{1}, and A_{2} will denote additive number systems. The most desirable of the additive systems are those which are reduced.

Definition 12 ([6], 2.41). For f : ℕ → ℕ the support of f is

supp f(n) = {n ∈ ℕ : f(n) > 0}.

Definition 13 ([6], 2.43). A is reduced if gcd(supp p(n)) = 1.

Notice that A_{1} ∗ A_{2} may be reduced even when A_{1} and A_{2} are not. However in this subsection we will only consider reduced systems A_{i}.

Definition 14 ([6], 2.12). Given A, for each B ⊆ A the generating series of B is

B(x) = ∑_{n≥0} b(n)x^{n}.

Definition 15 ([6], 1.24 and 3.18). Given ρ ≥ 0, a real-valued function f(n) that is eventually defined on ℕ and eventually positive is in RT_{ρ} if

lim_{n→∞} \frac{f(n − 1)}{f(n)} = ρ,

that is, the radius of convergence of ∑ f(n)x^{n} can be found by the ratio test.

A power series is in RT_{ρ} if the sequence of coefficients is in RT_{ρ}. A is in RT_{ρ} if a(n) ∈ RT_{ρ}, which can only occur if a(n) is eventually nonzero.

Definition 16 ([6], 3.1). For B ⊆ A, the asymptotic density δ(B) of B is as follows, provided the limit exists:

δ(B) = lim_{\begin{matrix}n→∞ \\ a(n)≠0 \end{matrix}} \frac{b(n)}{a(n)}.

We will use the following notational conventions.

The radius of convergence of the generating function A_{i}(x) of A_{i} is ρ_{i}, the counting functions are a_{i}(n) and p_{i}(n), and when defined δ_{i}(B_{i}) is the asymptotic density of B_{i} with respect to A_{i} for B_{i} ⊆ A_{i}. For A the preceding apply with the removal of the subscript i.

Since P, P_{1}, and P_{2} are nonempty by definition, we know that ρ, ρ_{1}, and ρ_{2} are in [0, ∞). From Lemma 2.23 of [6] we know that ρ, ρ_{1}, and ρ_{2} are in [0, 1].

Lemma 17. Let A = A_{1} ∗ A_{2} and B = B_{1} × B_{2} where B_{i} ⊆ A_{i}. Then: 1. B(x) = B_{1}(x) · B_{2}(x).

2. ∑_{n≥0} B(n)x^{n} = ∑_{n≥0} B_{1}(n)x^{n} · B_{2}(x).

Proof. Both items follow from Proposition 3.33 of [6]. For the second item notice that B(n) = ∑_{k≤n} B_{1}(k)b_{2}(n−k) is the same relation as that which holds between b(n), b_{1}(n), and b_{2}(n). □

Lemma 18. If A = A_{1} ∗ A_{2} then ρ = min{ρ_{1}, ρ_{2}}.

Proof. ρ = min{ρ_{1}, ρ_{2}} by Lemma 17 since A_{1}(x) and A_{2}(x) are power series with nonnegative coefficients. □

Definition 19. A has Property I if all partition sets of A have asymptotic density. A has Property II if ρ > 0, A(ρ) = ∞, and a(n) ∈ RT_{ρ}.

Our goal is to analyse when Property I holds. Some results of Bell, Bell et al., Burris and Sárközy, and Warlimont follow; then we will proceed towards the main result of the additive subsection, Theorem 28.

Proposition 20 ([6], 3.28). If A has Property I then δ(P) = 0.

Proposition 21 ([3], [15], [4]). If δ(P) = 0 then ρ > 0 and A(ρ) = ∞.

Proposition 22 ([6], 3.30). A has Property I and is reduced implies a(n) ∈ RT_{ρ}.

Corollary 23. A has Property I and is reduced implies A has Property II.

Proposition 24. Let A_{1} and A_{2} have Property II and be reduced, and let A = A_{1} ∗ A_{2}. If ρ_{1} ≤ ρ_{2}, then:

• lim_{n→∞} \frac{a_{1}(n)}{a(n)} = \begin{cases}0 & if ρ_{1} = ρ_{2} \\ A_{2}(ρ_{1})^{−1} & if ρ_{1} < ρ_{2}. \end{cases}

• lim_{n→∞} \frac{a_{2}(n)}{a(n)} = 0.

Proof. For the second item take n large enough that a_{2}(n) > 0. Then for m < n

\frac{a(n)}{a_{2}(n)} = ∑_{k=0}^{n} a_{1}(k)\frac{a_{2}(n − k)}{a_{2}(n)} ≥ ∑_{k=0}^{m} a_{1}(k)\frac{a_{2}(n − k)}{a_{2}(n)}.

So

lim inf_{n→∞} \frac{a(n)}{a_{2}(n)} ≥ ∑_{k=0}^{m} a_{1}(k)ρ_{2}^{k}.

Now A_{1}(ρ_{2}) = ∞; so by taking the limit as m → ∞ we get the desired result.

For the first item if ρ_{1} = ρ_{2} apply the above proof with 1 and 2 interchanged; if ρ_{1} < ρ_{2} then apply Schur’s Theorem.^{2} □

lim_{n→∞} \frac{[x^{n}](S(x) · T(x))}{[x^{n}]T(x)} = S(ρ).

2By this is meant Schur’s Tauberian Theorem which can be found in [6], 3.42. It states: Let S(x) and T(x) be two power series such that, for some ρ ≥ 0, T(x) ∈ RT_{ρ}, and S(x) has radius of convergence greater than ρ. Then If S(ρ) > 0 then this can be expressed as [x^{n}](S(x) · T(x)) ∼ S(ρ) · [x^{n}]T(x).

Proposition 25. Let A_{1} and A_{2} have Property I and be reduced with ρ_{1} = ρ_{2}. Then A_{1} ∗ A_{2} has Property I and for B a partition set of A_{1} ∗ A_{2} and B_{ij} as in Lemma 11 we get

δ(B) = ∑_{j=1}^{k} δ_{1}(B_{1j})δ_{2}(B_{2j}).

Proof. Let A = A_{1} ∗ A_{2}.

First, consider partition sets of A of the form B = B_{1} × B_{2} where each B_{i} is a partition set of A_{i}.

From b_{i}(n)/a_{i}(n) → δ_{i}(B_{i}) it follows that for all ϵ > 0 there is an N such that for m, n ≥ N,

|b_{1}(m)b_{2}(n) − δ_{1}(B_{1})δ_{2}(B_{2})a_{1}(m)a_{2}(n)| ≤ ϵa_{1}(m)a_{2}(n).

Then, using Proposition 24, and the fact that a linear combination ∑ r_{i} o(a(n)) is o(a(n)), we have

|b(n) − δ_{1}(B_{1})δ_{2}(B_{2})a(n)|

= | ∑_{j=0}^{n} b_{1}(j)b_{2}(n − j) − δ_{1}(B_{1})δ_{2}(B_{2})a_{1}(j)a_{2}(n − j)|

≤ | ∑_{j=N}^{n−N} b_{1}(j)b_{2}(n − j) − δ_{1}(B_{1})δ_{2}(B_{2})a_{1}(j)a_{2}(n − j)|

+ | ∑_{j=0}^{N−1} b_{1}(j)b_{2}(n − j) − δ_{1}(B_{1})δ_{2}(B_{2})a_{1}(j)a_{2}(n − j)|

+ | ∑_{j=0}^{N−1} b_{1}(n − j)b_{2}(j) − δ_{1}(B_{1})δ_{2}(B_{2})a_{1}(n − j)a_{2}(j)|

≤ ϵa(n) + | ∑_{j=0}^{N−1} b_{1}(j)o(a(n)) − δ_{1}(B_{1})δ_{2}(B_{2})a_{1}(j)o(a(n))|

+ | ∑_{j=0}^{N−1} o(a(n))b_{2}(j) − δ_{1}(B_{1})δ_{2}(B_{2})o(a(n))a_{2}(j)|

= ϵa(n) + o(a(n)).

Therefore δ(B) = δ_{1}(B_{1})δ_{2}(B_{2}).

Thus for a general partition set B of A

δ(B) = ∑_{j=1}^{k} δ_{1}(B_{1j})δ_{2}(B_{2j})

by Lemma 11 and Lemma 3.2 from [6]. Therefore A has Property I. □

Lemma 26. If lim_{n→∞} c(n)/d(n) = ℓ > 0 and d(n) ∈ RT_{ρ} then c(n) ∈ RT_{ρ}.

Proof. The hypotheses imply c(n) is eventually nonzero, and then

lim_{n→∞} \frac{c(n − 1)}{c(n)} = lim_{n→∞} \frac{c(n − 1)}{d(n − 1)} \frac{d(n)}{c(n)} \frac{d(n − 1)}{d(n)} = ρ.

□

Proposition 27. Let A_{1} and A_{2} have Property I and be reduced with ρ_{1} < ρ_{2}. Then A_{1} ∗ A_{2} has Property I. Let B be a partition set of A_{1} ∗ A_{2} and let B_{ij} be as in Lemma 11. Then

δ(B) = ∑_{j=1}^{k} δ_{1}(B_{1j})\frac{B_{2j}(ρ_{1})}{A_{2}(ρ_{1})}.

Proof. Let A = A_{1} ∗ A_{2}. We know A_{1} ∈ RT_{ρ} by Proposition 22. Thus A ∈ RT_{ρ} by Proposition 3.44 of [6].

Consider partition sets of A of the form B = B_{1} × B_{2} where B_{i} is a partition set of A_{i}, i = 1, 2.

First, suppose δ_{1}(B_{1}) ≠ 0. Then B_{1}(x) ∈ RT_{ρ} by Lemma 26. Thus by Lemma 17 we can apply Schur’s Theorem to get b(n) ∼ b_{1}(n)B_{2}(ρ_{1}) and a(n) ∼ a_{1}(n)A_{2}(ρ_{1}). Hence

δ(B) = δ_{1}(B_{1})\frac{B_{2}(ρ_{1})}{A_{2}(ρ_{1})}.

Second, suppose δ_{1}(B_{1}) = 0; then for all ϵ > 0 there exists an N such that b_{1}(n)/a_{1}(n) < ϵ for n ≥ N. Thus, since b_{2}(n) ≤ a_{2}(n) for all n,

b(n) = ∑_{j=0}^{n} b_{1}(n − j)b_{2}(j)

≤ ∑_{j=0}^{n−N} b_{1}(n − j)a_{2}(j) + ∑_{j=0}^{N−1} b_{1}(j)o(a(n)) by Proposition 24

≤ ϵ ∑_{j=0}^{n−N} a_{1}(n − j)a_{2}(j) + o(a(n))
≤ ϵa(n) + o(a(n))

which gives δ(B) = 0. Therefore in both cases δ(B) = δ_{1}(B_{1})\frac{B_{2}(ρ_{1})}{A_{2}(ρ_{1})}.

Hence for a general partition set B of A

δ(B) = ∑_{j=1}^{k} δ_{1}(B_{1j})\frac{B_{2j}(ρ_{1})}{A_{2}(ρ_{1})}

by Lemma 11 and Lemma 3.2 from [6]. Therefore A has Property I. □

Theorem 28. Let A_{1} and A_{2} have Property I and be reduced with ρ_{1} ≤ ρ_{2}. Then A_{1} ∗A_{2} has Property I and for B a partition set of A_{1} ∗A_{2} and B_{ij} as in Lemma 11 we get

δ(B) = \begin{cases}∑_{j=1}^{k} δ_{1}(B_{1j})\frac{B_{2j}(ρ_{1})}{A_{2}(ρ_{1})} & if ρ_{1} < ρ_{2}, \\ ∑_{j=1}^{k} δ_{1}(B_{1j})δ_{2}(B_{2j}) & if ρ_{1} = ρ_{2}. \end{cases}

Proof. If ρ_{1} = ρ_{2} apply Proposition 25; otherwise apply Proposition 27. □

Remark 29. If we assume only that A_{1} ∗ A_{2} has Property I then we can not conclude that A_{1} and A_{2} have Property I. For instance if 1 < q_{1} < q_{2}, where q_{2} is an integer, and we take A_{1} to be the additive number system with p_{1}(n) = ⌊q_{1}^{n}/n^{2}⌋ and A_{2} to be the additive number system with p_{2}(n) = q_{2}^{n}, then p(n) = q_{2}^{n} + ⌊q_{1}^{n}/n^{2}⌋ = q_{2}^{n} + O(q_{1}^{n}); so by Theorem 5.17 of [6], based on the asymptotics of Knopfmacher, Knopfmacher, and Warlimont, A_{1} ∗ A_{2} has Property I. However ρ_{1} = 1/q_{1} and P_{1}(1/q_{1}) < ∞ which gives A_{1}(1/q_{1}) < ∞; so A_{1} does not have Property I.^{3}

3.2. Multiplicative. Throughout the multiplicative subsections of this paper, unless otherwise specified, A, A_{1}, and A_{2} will denote multiplicative number systems.

Similarly to the additive situation, the most desirable multiplicative systems are those which are strictly multiplicative.

Definition 30 ([6], 9.39 and 9.48). A is discrete if there is a positive integer λ such that ∥a∥ is an integer power of λ for any a ∈ A. A is strictly multiplicative if it is not discrete.

Strictly multiplicative systems will be the focus of this subsection.

Definition 31 ([6], 8.6). Given A, for each set B ⊆ A the generating series of B is the Dirichlet series:

B(x) = ∑_{n≥1} b(n)n^{−x}.

Definition 32 ([6], 7.19 and 9.16). Given α ∈ ℝ, a function f(x) that is eventually defined on ℝ, and eventually positive, is in RV_{α} if for all y > 0

lim_{x→∞} \frac{f(xy)}{f(x)} = y^{α},

that is, f(x) has regular variation at infinity of index α. A is in RV_{α} if A(x) ∈ RV_{α}.

Definition 33 ([6], 9.3). For B ⊆ A, the global asymptotic density Δ(B) of B is as follows, provided the limit exists:

Δ(B) = lim_{n→∞} \frac{B(n)}{A(n)}.

We will use the following notational conventions.

The abscissa of convergence of the generating function A_{i}(x) of A_{i} is α_{i}, the functions a_{i}(n), p_{i}(n), and A_{i}(x) are the local and global counting functions, and when defined Δ_{i}(B_{i}) is the global asymptotic density of B_{i} with respect to A_{i} for B_{i} ⊆ A_{i}. For A the preceding apply with the removal of the subscript i.

3This counterexample is inspired by one suggested by the referee.

Since P, P_{1}, and P_{2} are nonempty by definition, we know that α, α_{1}, and α_{2} are nonnegative.

Lemma 34. Let A = A_{1} ∗ A_{2} and B = B_{1} × B_{2} where B_{i} ⊆ A_{i}. Then

B(x) = B_{1}(x) · B_{2}(x).

Proof. Apply Proposition 9.30 from [6]. □

Lemma 35. If A = A_{1} ∗ A_{2} then α = max{α_{1}, α_{2}}.

Proof. This follows from Lemma 34 since A_{1}(x) and A_{2}(x) both have nonnegative coefficients. □

Definition 36. A has Property I means that all partition sets of A have global asymptotic density. A has Property II if α < ∞, A(α) = ∞, and A ∈ RV_{α}.

As in the additive case Property I is the central property of interest. Some important results of Burris and Sárközy, and Warlimont follow; then we will proceed towards the main result of the multiplicative subsection, Theorem 47. Proposition 39 is a recently announced result of Warlimont proving conjecture 9.70 from [6].

Proposition 37 ([6], 9.28). If A has Property I then Δ(P) = 0.

Proposition 38 ([14]). If Δ(P) = 0 then α < ∞.

Proposition 39 ([16]). If Δ(P) = 0 then A(α) = ∞.

Proposition 40 ([6], 9.50). If A is strictly multiplicative and has Property I then A ∈ RV_{α}.

Corollary 41. A has Property I and is strictly multiplicative implies A has Property II.

We also need a multiplicative analogue of Schur’s Theorem:

Proposition 42 ([6, 9.53]). Given α ∈ ℝ, suppose S(x) and T(x) are two Dirichlet series such that T(x) has nonnegative coefficients, T (x) ∈ RV_{α}, and the abscissa of absolute convergence of S(x) is less than α. Let R(x) = S(x) · T(x). Then

lim_{x→∞} \frac{R(x)}{T (x)} = S(α).

If S(α) > 0 then this can be expressed as R(x) ∼ S(α) · T (x).

Proposition 43. Let A_{1} and A_{2} have Property II and be strictly multiplicative, and let A = A_{1} ∗ A_{2}. If α_{1} ≥ α_{2}, then:

• lim_{n→∞} \frac{A_{1}(n)}{A(n)} = \begin{cases}0 & if α_{1} = α_{2} \\ A_{2}(α_{1})^{−1} & if α_{1} > α_{2}. \end{cases}

• lim_{n→∞} \frac{A_{2}(n)}{A(n)} = 0.

Proof.

\frac{A(n)}{A_{2}(n)} = ∑_{i≤n} a_{1}(i)\frac{A_{2}(n/i)}{A_{2}(n)} ≥ ∑_{i≤m} a_{1}(i)\frac{A_{2}(n/i)}{A_{2}(n)}

for fixed m < n. So

lim inf_{n→∞} \frac{A(n)}{A_{2}(n)} ≥ ∑_{i≤m} a_{1}(i)i^{−α_{2}}.

Now A_{1}(α_{2}) = ∞; so by taking the limit as m → ∞ we get the second item.

For the first item, if α_{1} = α_{2} apply the above proof with 1 and 2 interchanged; if α_{1} > α_{2} then apply Proposition 42. □

Proposition 44. Let A_{1} and A_{2} have Property I and be strictly multiplicative with α_{1} = α_{2}. Then A_{1} ∗ A_{2} has Property I. Let B be a partition set of A_{1} ∗ A_{2} and let B_{ij} be as in Lemma 11. Then

Δ(B) = ∑_{j=1}^{k} Δ_{1}(B_{1j})Δ_{2}(B_{2j}).

Proof. Let A = A_{1} ∗ A_{2}.

First, consider partition sets of A of the form B = B_{1} × B_{2} where each B_{i} is a partition set of A_{i}.

Take an ϵ > 0. Let N be such that |B_{i}(n) − Δ_{i}(B_{i})A_{i}(n)| < ϵA_{i}(n) for n ≥ N and i = 1, 2. Notice that

∑_{j=1}^{n} b_{1}(j)A_{2}(n/j) = ∑_{j=1}^{n} B_{1}(n/j)a_{2}(j), (1)

since both sides are equal to the global counting function of B_{1} × A_{2}. Then

|B(n) − Δ_{1}(B_{1})Δ_{2}(B_{2})A(n)|

= | ∑_{j=1}^{n} b_{1}(j)B_{2}(n/j) − Δ_{1}(B_{1})Δ_{2}(B_{2})a_{1}(j)A_{2}(n/j)|

≤ | ∑_{j=1}^{n} b_{1}(j)B_{2}(n/j) − Δ_{2}(B_{2})b_{1}(j)A_{2}(n/j)|

+ | ∑_{j=1}^{n} Δ_{2}(B_{2})b_{1}(j)A_{2}(n/j) − Δ_{1}(B_{1})Δ_{2}(B_{2})a_{1}(j)A_{2}(n/j)|

= \underbrace{| ∑_{j=1}^{n} b_{1}(j)B_{2}(n/j) − Δ_{2}(B_{2})b_{1}(j)A_{2}(n/j)|}_{I}

+ Δ_{2}(B_{2}) \underbrace{| ∑_{j=1}^{n} B_{1}(n/j)a_{2}(j) − Δ_{1}(B_{1})A_{1}(n/j)a_{2}(j)|}_{II} by (1).

Applying the triangle equality gives

Term I ≤ ∑_{1≤j≤n/N} b_{1}(j)|B_{2}(n/j) − Δ_{2}(B_{2})A_{2}(n/j)|

+ ∑_{n/N<j≤n} b_{1}(j)|B_{2}(n/j) − Δ_{2}(B_{2})A_{2}(n/j)|

≤ ϵ ∑_{1≤j≤n/N} b_{1}(j)A_{2}(n/j)

+ ∑_{1≤k<N} ∑_{\frac{n}{k+1} <j≤ \frac{n}{k}} b_{1}(j)|B_{2}(k) − Δ_{2}(B_{2})A_{2}(k)|

≤ ϵA(n) + ∑_{k=1}^{N−1} (B_{1} (\frac{n}{k}) − B_{1}(\frac{n}{k+1}))|B_{2}(k) − Δ_{2}(B_{2})A_{2}(k)|

= ϵA(n) + o(A(n)) by Proposition 43.

Term II can be treated in the same way. Therefore Δ(B) = Δ_{1}(B_{1})Δ_{2}(B_{2}). Thus for a general partition set B of A

Δ(B) = ∑_{j=1}^{k} Δ_{1}(B_{1j})Δ_{2}(B_{2j})

by Lemma 11 and Lemma 9.4 from [6]. Therefore A has Property I. □

Lemma 45. Let B be a partition set of A. If Δ(B) ≠ 0 and A(x) ∈ RV_{α} then B(x) ∈ RV_{α}.

Proof. B(x) is eventually nonzero, since Δ(B) ≠ 0. Then, for all y > 0,

lim_{x→∞} \frac{B(xy)}{B(x)} = lim_{x→∞} \frac{B(xy)}{A(xy)} \frac{A(x)}{B(x)} \frac{A(xy)}{A(x)}
= lim_{x→∞} \frac{B(⌊xy⌋)}{A(⌊xy⌋)} \frac{A(⌊x⌋)}{B(⌊x⌋)} \frac{A(xy)}{A(x)} = y^{α}.
□

Proposition 46. Let A_{1} and A_{2} have Property I and be strictly multiplicative with α_{1} > α_{2}. Then A_{1} ∗ A_{2} has Property I. Also for B a partition set of A_{1} ∗ A_{2} and for B_{ij} as in Lemma 11 we get

Δ(B) = ∑_{j=1}^{k} Δ_{1}(B_{1j})\frac{B_{2j}(α_{1})}{A_{2}(α_{1})}.

Proof. Let A = A_{1} ∗ A_{2}.

Consider partition sets of A of the form B = B_{1} × B_{2} where B_{i} is a partition set of A_{i}, i = 1, 2.

First, suppose Δ_{1}(B_{1}) ≠ 0. Then B_{1}(x) ∈ RV_{α_{1}} by Lemma 45. Thus by Lemma 34 we can apply Proposition 42 to get B(x) ∼ B_{1}(x)B_{2}(α_{1}) and A(x) ∼ A_{1}(x)A_{2}(α_{1}). So, as n → ∞,

\frac{B(n)}{A(n)} → Δ_{1}(B_{1})\frac{B_{2}(α_{1})}{A_{2}(α_{1})}.

Second, suppose Δ_{1}(B_{1}) = 0. Then for all ϵ > 0 there is an N such that B_{1}(n)/A_{1}(n) < ϵ for n ≥ N. Thus

B(n) = ∑_{1≤j≤n/N} B_{1}(n/j)b_{2}(j) + ∑_{n/N<j≤n} B_{1}(n/j)b_{2}(j)
≤ ∑_{1≤j≤n/N} B_{1}(n/j)b_{2}(j) + ∑_{1≤k<N} ∑_{\frac{n}{k+1} <j≤ \frac{n}{k}} B_{1}(k)b_{2}(j)

≤ ϵ ∑_{1≤j≤n/N} A_{1}(n/j)a_{2}(j) + ∑_{1≤k<N} B_{1}(k)(B_{2}(\frac{n}{k}) − B_{2}(\frac{n}{k+1}))
≤ ϵA(n) + ∑_{1≤k<N} B_{1}(k)o(A(n))  by Proposition 43
≤ ϵA(n) + o(A(n))

which gives Δ(B) = 0. Therefore in both cases Δ(B) = Δ_{1}(B_{1})\frac{B_{2}(α_{1})}{A_{2}(α_{1})}.

Hence for a general partition set B of A

Δ(B) = ∑_{j=1}^{k} Δ_{1}(B_{1j})\frac{B_{2j}(α_{1})}{A_{2}(α_{1})}

by Lemma 11 and Lemma 9.4 from [6]. Therefore A has Property I. □

Theorem 47. Let A_{1} and A_{2} have Property I and be strictly multiplicative with α_{1} ≥ α_{2}. Then A = A_{1} ∗ A_{2} has Property I. Also for B a partition set of A_{1} ∗ A_{2} and for B_{ij} as in Lemma 11 we get

Δ(B) = \begin{cases}∑_{j=1}^{k} Δ_{1}(B_{1j})\frac{B_{2j}(α_{1})}{A_{2}(α_{1})} & if α_{1} > α_{2}, \\ ∑_{j=1}^{k} Δ_{1}(B_{1j})Δ_{2}(B_{2j}) & if α_{1} = α_{2}. \end{cases}

Proof. If α_{1} = α_{2} apply Proposition 44; otherwise apply Proposition 46. □

4. The general situation

4.1. Additive. Now, returning to the additive notational conventions, we extend the results to the case when A_{1} and A_{2} are not necessarily reduced.

Definition 48. Let d = gcd(supp p(n)) and let j|d. Define A^{∗_{j}} to be the additive number system obtained from A by altering the norm so that ∥a∥^{∗_{j}} = ∥a∥/j. Let a^{∗_{j}}(n) and p^{∗_{j}}(n) be the counting functions of A^{∗_{j}}, and let ρ^{∗_{j}} be the radius of convergence of A^{∗_{j}}(x), the generating function of A^{∗_{j}}.

Notice that a^{∗_{j}}(n) = a(nj), p^{∗_{j}}(n) = p(nj), gcd(supp p^{∗_{j}}(n)) = 1, and ρ^{∗_{j}} = ρ^{j}. If d = j then we drop the j in the exponent and write A^{∗}, which is a reduced additive system called the reduced form of A. Notice also that if A is not reduced then A ∉ RT_{ρ} since a(n) is infinitely often zero.

As an additional notational convention we will let d_{i} = gcd(supp p_{i}(n)) and d = gcd(supp p(n)).

Here is an example of the value of working with nonreduced systems. For q a power of a prime and d a positive integer, define the additive number system A_{q,d} as the system formed from the monic elements of 𝔽_{q}[x^{d}] with polynomial multiplication as the operation and ∥g(x)∥ = deg g(x).

Let a_{q,d}(n) be the local counting function for A_{q,d}. Then

a_{q,d}(n) = \begin{cases}q^{m} & if dm = n \\ 0 & if d ∤ n. \end{cases}

So the radius of convergence of the generating function of A_{q,d} is

ρ = \frac{1}{q^{1/d}} .

Now, for all d, A_{q,d}^{∗} looks like A_{q,1} under the mapping g(x^{d}) ↦ g(x). From a_{q,1}(n) = q^{n} we see by the Knopfmacher, Knopfmacher, and Warlimont asymptotics (see Theorem 5.17, [6]) that A_{q,1} has Property I and so, as we will see in Lemma 51, A_{q,d} has Property I for all d.

Determining when A_{q_{1},d_{1}} ∗ A_{q_{2},d_{2}} has Property I is rather more involved.

Theorem 57, the main theorem of this subsection, shows that A_{q_{1},d_{1}} ∗ A_{q_{2},d_{2}} has Property I iff

• 1/q_{1}^{1/d_{1}} < 1/q_{2}^{1/d_{2}} and d_{1}|d_{2}, or

• 1/q_{1}^{1/d_{1}} > 1/q_{2}^{1/d_{2}} and d_{2}|d_{1}, or

• 1/q_{1}^{1/d_{1}} = 1/q_{2}^{1/d_{2}}.

For instance, A_{2,2} ∗ A_{2,4} and A_{2,1} ∗ A_{2,3} have Property I but A_{2,2} ∗ A_{2,3} does not. Such facts require a careful study of nonreduced systems.

Lemma 49. For all B ⊆ A, δ(B) = lim_{n→∞} \frac{b^{∗}(n)}{a^{∗}(n)}, provided δ(B) exists.

Proof. From Hua’s Theorem (as found for instance in [6], 2.49), a^{∗}(n) is eventually

nonzero; so lim_{n→∞} \frac{b^{∗}(n)}{a^{∗}(n)} = lim_{\begin{matrix}n→∞ \\ a(n)≠0 \end{matrix}} \frac{b(n)}{a(n)}. □

The next lemma follows easily from the definitions.

Lemma 50. Let A = A_{1} ∗ A_{2}; then d = gcd(d_{1}, d_{2}). Let j|d; then A^{∗_{j}} = A_{1}^{∗_{j}} ∗ A_{2}^{∗_{j}} iff A = A_{1} ∗ A_{2}.

Lemma 51. A^{∗_{j}} has Property I iff A has Property I, whenever j|d.

Proof. A and A^{∗_{j}} have the same reduced form, so apply Lemma 3.3 from [6]. □

From this we get a slight variation on Proposition 22.

Proposition 52. A has Property I implies a^{∗}(n) ∈ RT_{ρ^{∗}}.

The key to the more general case is Proposition 54, a modified version of Proposition 24. In order to prove Proposition 54 we will need the following Lemma.

Lemma 53. If f(n) ∈ RT_{ρ}, ρ > 0, and ∑_{n≥0} f(n)ρ^{n} = ∞ then for all k and all

m ≥ 1 we have ∑_{n≡k mod m} f(n)ρ^{n} = ∞.

Proof. Pick an ϵ > 0. Let N be such that f(j + 1) > 0 and f(j)/f(j + 1) < ϵ + ρ for j > N. Then, for suitable constants C_{1} and C,

∑_{j≡k mod m} f(j)ρ^{j} = ∑_{\begin{matrix}j≡k mod m \\ j≤N \end{matrix}} f(j)ρ^{j} + ∑_{\begin{matrix}j≡k mod m \\ j>N \end{matrix}} f(j)ρ^{j}

≤ C_{1} + (ϵ + ρ) ∑_{\begin{matrix}j≡k mod m \\ j>N \end{matrix}} f(j + 1)ρ^{j}

= C + (\frac{ϵ + ρ}{ρ}) ∑_{j≡k+1 mod m} f(j)ρ^{j}.

Now ∑_{n≥0} f(n)ρ^{n} = ∞ implies that for all m we have ∑_{j≡k mod m} f(j)ρ^{j} = ∞ for

some k. Then by the above the latter equation holds for all k □

Proposition 54. Let A = A_{1} ∗ A_{2}, gcd(d_{1}, d_{2}) = 1, A_{1}(ρ_{1}) = ∞, and A_{2}^{∗} ∈ RT_{ρ_{2}^{∗}}. If ρ_{1} ≤ ρ_{2} then lim_{n→∞} a_{2}(n − j)/a(n) = 0 for all integers j ≥ 0.

Proof. Let us restrict out attention to n large enough that a(n − j) > 0. This is possible since d = gcd(d_{1}, d_{2}) = 1; so A is reduced. For d_{2} ∤ n − j we have a_{2}(n − j) = 0, and so a_{2}(n − j)/a(n) = 0. Assume d_{2}|n − j, and let d_{2}m = n − j. Now

\frac{a(d_{2}m + j)}{a_{2}(d_{2}m)} = ∑_{k=0}^{d_{2}m+j} a_{1}(k)\frac{a_{2}(d_{2}m + j − k)}{a_{2}(d_{2}m)}

= ∑_{−j/d_{2}≤i≤m} a_{1}(d_{2}i + j)\frac{a_{2}(d_{2}m − d_{2}i)}{a_{2}(d_{2}m)}

= ∑_{−j/d_{2}≤i≤m} a_{1}(d_{2}i + j)\frac{a_{2}^{∗}(m − i)}{a_{2}^{∗}(m)}

≥ ∑_{i=0}^{k} a_{1}(d_{2}i + j)\frac{a_{2}^{∗}(m − i)}{a_{2}^{∗}(m)}

for fixed k < m. Thus

lim inf_{m→∞} \frac{a(d_{2}m + j)}{a_{2}(d_{2}m)} ≥ ∑_{i=0}^{k} a_{1}(d_{2}i + j) lim inf_{m→∞} \frac{a_{2}^{∗}(m − i)}{a_{2}^{∗}(m)}

= ∑_{i=0}^{k} a_{1}(d_{2}i + j)ρ_{2}^{d_{2}i}.

Taking the limit as k → ∞, we have

lim inf_{m→∞} \frac{a(d_{2}m + j)}{a_{2}(d_{2}m)} ≥ ∑_{i=0}^{∞} a_{1}(d_{2}i + j)ρ_{2}^{d_{2}i}
= ρ_{2}^{−j} ∑_{d_{1}ℓ≡j mod d_{2}} a_{1}(d_{1}ℓ)ρ_{2}^{d_{1}ℓ}
≥ ρ_{2}^{−j} ∑_{d_{1}ℓ≡j mod d_{2}} a_{1}^{∗}(ℓ)(ρ_{1}^{∗})^{ℓ}

since ρ_{2}^{d_{1}} ≥ ρ_{1}^{d_{1}} = ρ_{1}^{∗}. Now since gcd(d_{1}, d_{2}) = 1, d_{1}ℓ ≡ j mod d_{2} is the same as ℓ ≡ c mod d_{2} for some c. We know that ∑_{ℓ≥0} a_{1}^{∗}(ℓ)(ρ_{1}^{∗})^{ℓ} = ∞, and ρ_{1} > 0 since A_{1}(ρ_{1}) = ∞. Apply Lemma 53 to get lim_{n→∞} a_{2}(n − j)/a(n) = 0. □

Proposition 55. Let A_{1} and A_{2} have Property I with ρ_{1} = ρ_{2}. Then A_{1} ∗ A_{2} has Property I. Let B be a partition set of A_{1} ∗A_{2} and let B_{ij} be as in Lemma 11. Then

δ(B) = ∑_{j=1}^{k} δ_{1}(B_{1j})δ_{2}(B_{2j}).

Proof. Let A = A_{1} ∗ A_{2}. By Lemmas 50 and 51 we need only consider d = 1. Since ρ_{1} = ρ_{2} we can switch the roles of A_{1} and A_{2} in Proposition 54 to get a_{1}(n − j) = o(a(n)) for all integers j ≥ 0.

From b_{i}(nd_{i})/a_{i}(nd_{i}) → δ_{i}(B_{i}) it follows that for all ϵ > 0 there is an N such that for md_{1}, nd_{2} ≥ N,

|b_{1}(md_{1})b_{2}(nd_{2}) − δ_{1}(B_{1})δ_{2}(B_{2})a_{1}(md_{1})a_{2}(nd_{2})| < ϵa_{1}(md_{1})a_{2}(nd_{2}).

Clearly a_{i}(k) = 0 gives b_{i}(k) = 0 for all k; so for j, k ≥ N,

|b_{1}(j)b_{2}(k) − δ_{1}(B_{1})δ_{2}(B_{2})a_{1}(j)a_{2}(k)| ≤ ϵa_{1}(j)a_{2}(k).

Now continue as in the proof of Proposition 25 using Proposition 54 in place of Proposition 24. □

Proposition 56. Let A_{1} and A_{2} have Property I with ρ_{1} < ρ_{2}. Then A_{1} ∗ A_{2} has Property I iff d_{1}|d_{2}. If d_{1}|d_{2} then for B a partition set of A_{1} ∗ A_{2} and for B_{ij} as in Lemma 11 we get

δ(B) = ∑_{j=1}^{k} δ_{1}(B_{1j})\frac{B_{2j}(ρ_{1})}{A_{2}(ρ_{1})}.

Proof. Let A = A_{1} ∗ A_{2}. d_{1}|d_{2} iff d_{1} = d since d = gcd(d_{1}, d_{2}). We need only consider the case when d = 1, that is, A is reduced, because d_{1}|d_{2} iff \frac{d_{1}}{d} |\frac{d_{2}}{d} and Lemmas 50 and 51 give A_{1} ∗ A_{2} has Property I iff A_{1}^{∗_{d}} ∗ A_{2}^{∗_{d}} has Property I. A_{1} ∈ RT_{ρ} iff A ∈ RT_{ρ} by Proposition 3.44 of [6].

(⇒) Suppose that A has Property I and radius of convergence ρ. Then A ∈ RT_{ρ} since A is reduced, and so A_{1} ∈ RT_{ρ}. Therefore A_{1} is reduced, and so d_{1} = 1 = d.

(⇐) Suppose that d_{1} = 1 = d; then A_{1} ∈ RT_{ρ_{1}}. Now continue as in the proof of Proposition 27 using Proposition 54 in place of Proposition 24. □

Theorem 57. Let A_{1} and A_{2} have Property I with ρ_{1} ≤ ρ_{2}. Then A_{1} ∗ A_{2} has Property I iff d_{1}|d_{2} or ρ_{1} = ρ_{2}. If A_{1} ∗ A_{2} has Property I then for B a partition set of A_{1} ∗ A_{2} and B_{ij} as in Lemma 11 we get

δ(B) = \begin{cases}∑_{j=1}^{k} δ_{1}(B_{1j})\frac{B_{2j}(ρ_{1})}{A_{2}(ρ_{1})} & if ρ_{1} < ρ_{2}, \\ ∑_{j=1}^{k} δ_{1}(B_{1j})δ_{2}(B_{2j}) & if ρ_{1} = ρ_{2}. \end{cases}

Proof. If ρ_{1} = ρ_{2} apply Proposition 55; otherwise apply Proposition 56. □

For any additive number system A define A_{q}, the extension of A by an indecomposable q, to be the additive system formed by adding a new indecomposable q to P, by letting A_{q} be the set of formal expressions q^{m} ∗ a, for m a nonnegative integer and a ∈ A, and by extending the norm with the definition ∥q^{m} ∗ a∥ = m∥q∥ + ∥a∥.

Corollary 58. Suppose A has Property I. An extension A_{q} has Property I iff d|∥q∥ or ρ = 1.

Proof. We can form an additive number system Q = (A_{Q}, P_{Q}, ∗, e, ∥ ∥) generated by q by letting P_{Q} = {q}, A_{Q} = {q^{m} : m ≥ 0}, q^{m_{1}} ∗ q^{m_{2}} = q^{m_{1}+m_{2}}, e = q^{0}, and ∥q^{m}∥ = m∥q∥. Then A_{q} ≅ A ∗ Q, Q has Property I, ρ_{Q} = 1, and d_{Q} = ∥q∥. Now apply Theorem 57. □

If A is reduced in the above corollary then we have Theorem 3.59 of [6].

Corollary 59. Let A_{1}, …, A_{n} be additive number systems with A_{1}^{∗}, …, A_{n}^{∗} in RT_{1}. Then (A_{1} ∗ ⋯ ∗ A_{n})^{∗} is also in RT_{1}.

Proof. Let A = A_{1} ∗ ⋯ ∗ A_{n}. It suffices to prove the theorem for n = 2. By Theorem 4.2 of [6] A_{i}^{∗} ∈ RT_{1} iff A_{i}^{∗} has Property I and ρ_{i} = 1, and likewise for A^{∗}. Now apply Theorem 57. □

This gives the additive half of Theorem 16.1 from [7]. If all the A_{i} are reduced then we have Stewart’s Theorem ([6], 4.15). We can also use Theorem 57 to prove the following corollary, extending a result of Bateman and Erdős which in our notation says: if p(n) ≤ 1, for n ≥ 1, then A^{∗} ∈ RT_{1}, ([6], 4.13).

Corollary 60. If p(n) ≤ c, for n ≥ 1, then A^{∗} ∈ RT_{1}

Proof. Let m = max{p(n) : n ≥ 1}, which exists since p(n) ≤ c is integer valued. Then we can construct additive number systems A_{i}, 1 ≤ i ≤ m, such that p_{i}(n) ≤ 1 for 1 ≤ i ≤ m and A = A_{1} ∗ ⋯ ∗ A_{m}. By the Bateman and Erdős result each A_{i}^{∗} ∈ RT_{1}; so by Corollary 59 A^{∗} ∈ RT_{1}. □

4.2. Multiplicative. Now, using the multiplicative notational conventions, we extend the results to the case when A_{1} and A_{2} may be discrete.

As an additional notational convention, if A is discrete λ will denote the largest integer such that ∥a∥ is an integer power of λ for all a ∈ A; and similarly for λ_{i} when A_{i} is discrete.

Although A having Property I does not imply A ∈ RV_{α} the following lemma gives a weaker result of a similar flavour.

Lemma 61. For any multiplicative number system A with Property I and abscissa

of convergence α, there exists a C > 0 such that lim inf_{n→∞} \frac{A(n/i)}{A(n)} ≥ Ci^{−α}, for all

integers i > 0.

Proof. If A is strictly multiplicative then, by Corollary 9.50 from [6], A ∈ RV_{α}, and so the lemma holds with C = 1.

Suppose A is discrete multiplicative; then Ã = (A, P, ∗, e, log_{λ} ∥ ∥) is a reduced additive number system with all partition sets having global asymptotic density, and with the radius of convergence of Ã(x) being λ^{−α}. Notice also that Ã(x) = A(λ^{x})

(Section 9.8.2 of [6] discusses relationships between A and Ã). Thus, by Proposition 3.20 and Lemma 3.29 from [6], Ã(n) ∈ RT_{λ^{−α}}; so

\frac{A(n/i)}{A(n)} = \frac{Ã(log_{λ}(n) − log_{λ}(i))}{Ã(log_{λ}(n))}

≥ \frac{Ã(⌊log_{λ}(n)⌋ − ⌈log_{λ}(i)⌉)}{Ã(⌊log_{λ}(n)⌋)}
→ λ^{−α⌈log_{λ} i⌉}.

Therefore

lim inf_{n→∞} \frac{A(n/i)}{A(n)} ≥ λ^{−α⌈log_{λ} i⌉} ≥ λ^{−α(log_{λ} i+1)} = λ^{−α}i^{−α}.

So let C = λ^{−α}. □

Proposition 62. Let A_{1} and A_{2} have Property I and let A = A_{1} ∗A_{2}. If α_{1} ≥ α_{2}, then:

• lim_{n→∞} \frac{A_{1}(n)}{A(n)} = \begin{cases}0 & if α_{1} = α_{2} \\ A_{2}(α_{1})^{−1} & if α_{1} > α_{2}. \end{cases}

• lim_{n→∞} \frac{A_{2}(n)}{A(n)} = 0.

Proof.

\frac{A(n)}{A_{2}(n)} = ∑_{i≤n} a_{1}(i)\frac{A_{2}(n/i)}{A_{2}(n)} ≥ ∑_{i≤m} a_{1}(i)\frac{A_{2}(n/i)}{A_{2}(n)}

for fixed m < n. So

lim inf_{n→∞} \frac{A(n)}{A_{2}(n)} ≥ lim inf_{n→∞} ∑_{i≤m} a_{1}(i)\frac{A_{2}(n/i)}{A_{2}(n)} ≥ C ∑_{i≤m} a_{1}(i)i^{−α}

by Lemma 61. By taking the limit as m → ∞ we get the second item.

For the first item, if α_{1} = α_{2} apply the above proof with 1 and 2 interchanged; if α_{1} > α_{2} then apply Proposition 42. □

Proposition 63. Let A_{1} and A_{2} have Property I with α_{1} = α_{2}. Then A_{1} ∗ A_{2} has Property I. Let B be a partition set of A_{1} ∗A_{2} and let B_{ij} be as in Lemma 11. Then

Δ(B) = ∑_{j=1}^{k} Δ_{1}(B_{1j})Δ_{2}(B_{2j}).

Proof. Follow the proof to Proposition 44 replacing references to Lemma 43 with references to Lemma 62. □

Proposition 64. Let A_{1} and A_{2} have Property I with α_{1} > α_{2}, and let A_{1} be strictly multiplicative. Then A_{1} ∗ A_{2} has Property I. Let B be a partition set of A_{1} ∗ A_{2} and let B_{ij} be as in Lemma 11. Then

Δ(B) = ∑_{j=1}^{k} Δ_{1}(B_{1j})\frac{B_{2j}(α_{1})}{A_{2}(α_{1})}.

Proof. Follow the proof to Proposition 46 replacing references to Lemma 43 with references to Lemma 62. □

For the purposes of the next lemma we will briefly escape from our multiplicative notational conventions. This lemma corresponds to Proposition 3.44 from [6], but is concerned with global counting functions rather than local counting functions.

Lemma 65. Suppose A = A_{1} ∗A_{2}, all three of which are additive number systems. Additionally suppose ∑_{n≥0} A_{1}(n)x^{n} has radius of convergence ρ and ∑_{n≥0} A_{2}(n)x^{n} has radius of convergence ρ_{2} > ρ. Then A(n) ∈ RT_{ρ} iff A_{1}(n) ∈ RT_{ρ}.

Proof. By Lemma 2.31 from [6] the radius of convergence of A_{2}(x) is also ρ_{2}. By Corollary 2.39 from [6] we have that [1/A_{2}](x) has radius of convergence at least ρ_{2}, where [1/A_{2}](x) is the power series expansion of 1/A_{2}(x). Then, by Lemma 17

∑_{n≥0} A(n)x^{n} = ∑_{n≥0} A_{1}(n)x^{n} · A_{2}(x)
∑_{n≥0} A_{1}(n)x^{n} = [1/A_{2}](x) · ∑_{n≥0} A(n)x^{n}.

It follows from Schur’s Theorem that A(n) ∈ RT_{ρ} iff A_{1}(n) ∈ RT_{ρ}. □

Proposition 66. Let A_{1} and A_{2} have Property I with α_{1} > α_{2}, and let A_{1} and A_{2} be discrete with λ_{2} a power of λ_{1}. Then A_{1} ∗ A_{2} has Property I. Let B be a partition set of A_{1} ∗ A_{2} and let B_{ij} be as in Lemma 11. Then

Δ(B) = ∑_{j=1}^{k} Δ_{1}(B_{1j})\frac{B_{2j}(α_{1})}{A_{2}(α_{1})}.

Proof. Let A = A_{1} ∗ A_{2}. From the hypotheses A is discrete with λ = λ_{1}. Therefore each Ã_{i} = (A_{i}, P_{i}, ∗, e, log_{λ} ∥ ∥_{i}) is an additive number system and Ã = (A, P, ∗, e, log_{λ} ∥ ∥) and Ã_{1} are reduced additive number systems.

Notice that Ã = Ã_{1} ∗ Ã_{2} since log_{λ} ∥(a_{1}, a_{2})∥ = log_{λ} ∥a_{1}∥_{1} + log_{λ} ∥a_{2}∥_{2} for all a_{1} ∈ A_{1} and a_{2} ∈ A_{2}. Also Ã_{1} and Ã_{2} have all partition sets with global asymptotic density since A_{1} and A_{2} do.

Consider partition sets of A of the form B = B_{1} × B_{2} where B_{i} is a partition set of A_{i}, i = 1, 2.

First, suppose Δ_{1}(B_{1}) ≠ 0. Then Δ_{1}(B̃_{1}) ≠ 0. So B̃_{1}(n) ∈ RT_{λ^{−α}} by Lemma 26. Thus B̃(n) ∼ B̃_{1}(n)B̃_{2}(λ^{−α_{1}}) and Ã(n) ∼ Ã_{1}(n)Ã_{2}(λ^{−α_{1}}) by Lemma 17 and Schur’s Theorem. So, as n → ∞,

\frac{B(n)}{A(n)} = \frac{B̃(⌊log_{λ} n⌋)}{Ã(⌊log_{λ} n⌋)} → Δ_{1}(B̃_{1})\frac{B̃_{2}(λ^{−α_{1}})}{Ã_{2}(λ^{−α_{1}})} = Δ_{1}(B_{1})\frac{B_{2}(α_{1})}{A_{2}(α_{1})}.

Second, suppose Δ_{1}(B_{1}) = 0. The proof that Δ(B) = 0 is exactly as in the proof

of Proposition 46. Therefore in both cases Δ(B) = Δ_{1}(B_{1})\frac{B_{2}(α_{1})}{A_{2}(α_{1})}.

Hence for a general partition set B of A

Δ(B) = ∑_{j=1}^{k} Δ_{1}(B_{1j})\frac{B_{2j}(α_{1})}{A_{2}(α_{1})}

by Lemma 11 and Lemma 9.4 from [6]. Therefore A has Property I. □

Proposition 67. If A_{1} ∗ A_{2} has Property I and α_{1} ≥ α_{2}, then • α_{1} = α_{2} or

• α_{1} > α_{2} and A_{1} is strictly multiplicative or

• α_{1} > α_{2}, A_{1} and A_{2} are discrete, and λ_{2} is a power of λ_{1}.

Proof. Let A = A_{1} ∗ A_{2}. Assume α_{1} > α_{2}.

Suppose A is strictly multiplicative; then A ∈ RV_{α} by Corollary 9.50 from [6]. Thus A_{1} ∈ RV_{α} by Proposition 9.55 from [6]. Then A_{1} is strictly multiplicative; for otherwise α_{1} = 0 by Proposition 9.51 from [6] which contradicts α_{1} > α_{2}.

Suppose A is discrete; then ∥(a_{1}, a_{2})∥ is an integer power of λ for a_{i} ∈ A_{i}. Thus if there existed an a_{1} ∈ A_{1} such that ∥a_{1}∥_{1} was not an integer power of λ then ∥(a_{1}, e)∥ = ∥a_{1}∥_{1} would not be an integer power of λ, and likewise for a_{2} ∈ A_{2}. Therefore for all a_{i} ∈ A_{i}, i = 1, 2, ∥a_{i}∥_{i} is an integer power of λ; so A_{1} and A_{2} are discrete. Hence λ_{1} and λ_{2} are powers of λ, each Ã_{i} = (A_{i}, P_{i}, ∗, e, log_{λ} ∥ ∥_{i}) is an additive number system, and Ã = (A, P, ∗, e, log_{λ} ∥ ∥) is a reduced additive number system.

As in Proposition 66 Ã = Ã_{1} ∗ Ã_{2} and Ã_{1} and Ã_{2} have all partition sets with global asymptotic density. So by Proposition 3.20 and Lemma 3.29 from [6], Ã(n) ∈ RT_{λ^{−α}}; so Ã_{1}(n) ∈ RT_{λ^{−α}} by Lemma 65. Now α = α_{1} > α_{2} ≥ 0; so λ^{−α} < 1. If Ã_{1} is not reduced then Ã_{1}(n − 1)/Ã_{1}(n) is infinitely often 1 giving a contradiction. So Ã_{1} is reduced, and thus λ = λ_{1}. Therefore λ_{2} is a power of λ_{1}. □

Theorem 68. Let A_{1} and A_{2} have Property I with α_{1} ≥ α_{2}. Then A = A_{1} ∗ A_{2} has Property I iff

• α_{1} = α_{2} or

• α_{1} > α_{2} and A_{1} is strictly multiplicative or

• α_{1} > α_{2}, A_{1} and A_{2} are discrete, and λ_{2} is a power of λ_{1}. If A_{1} ∗ A_{2} has Property I then for B a partition set of A_{1} ∗ A_{2} and for B_{ij} as in Lemma 11 we get

Δ(B) = \begin{cases}∑_{j=1}^{k} Δ_{1}(B_{1j})\frac{B_{2j}(α_{1})}{A_{2}(α_{1})} & if α_{1} > α_{2}, \\ ∑_{j=1}^{k} Δ_{1}(B_{1j})Δ_{2}(B_{2j}) & if α_{1} = α_{2}. \end{cases}

Proof. In the if direction apply Propositions 63, 64, and 66, and in the only if direction apply Proposition 67. □

Because of the connection between discrete multiplicative systems and additive systems used in Lemma 61 and Proposition 66, global density results for multiplicative systems give global density results for additive systems. Each additive system can be seen as a multiplicative system (as shown in Section 9.8.1 of [6]); this translation maps additive number systems in RT_{ρ} to multiplicative number systems in RV_{− log_{λ} ρ} for integer λ > 1 and preserves partition sets and density. In the next corollary we will also need the fact that if the generating series of the additive number system diverges at its radius of convergence then the Dirichlet series of the corresponding multiplicative number system diverges at its abscissa of convergence.

For the purposes of this corollary we are returning to the notational conventions of the first half of the paper.

Corollary 69. Let A_{1} and A_{2} be additive number systems such that every partition set has global asymptotic density. Then every partition set of A_{1} ∗ A_{2} has global asymptotic density iff d_{1}|d_{2} or ρ_{1} = ρ_{2}.

Proof. Let A = A_{1}∗A_{2}. Pick an integer λ_{0} > 1. Then Ã_{i} = (A_{i}, P_{i}, ∗, e, λ_{0}^{∥ ∥_{i}}) and Ã = (A, P, ∗, e, λ_{0}^{∥ ∥}) are discrete multiplicative number systems with Ã = Ã_{1} ∗ Ã_{2}. Apply Theorem 68. Notice that in the case ρ_{1} < ρ_{2} the condition d_{1}|d_{2} is equivalent to the condition λ_{2} is a power of λ_{1} for the discrete systems. □

Corollary 70. Let A_{1}, …, A_{n} be multiplicative number systems in RV_{0}. Then A_{1} ∗ ⋯ ∗ A_{n} ∈ RV_{0}.

Proof. It suffices to prove the theorem for n = 2. A_{i} ∈ RV_{0} iff A_{i} has Property I and the abscissa of convergence of A_{i} is 0 by Theorem 10.2 of [6]; the same applies to A_{1} ∗ A_{2}. Now apply Theorem 68. □

This gives Odlyzko’s Theorem ([6], 10.5) and the multiplicative half of Theorem 16.1 from [7].

For any multiplicative number system A define A_{q}, the extension of A by an indecomposable q, to be the multiplicative number system formed by adding a new indecomposable q to P, by letting A_{q} be the set of formal expressions q^{m} ∗ a, for m a nonnegative integer and a ∈ A, and by extending the norm by the definition ∥q^{m} ∗ a∥ = ∥q∥^{m}∥a∥.

Corollary 71. Suppose that A has Property I. An extension A_{q} has Property I iff A is strictly multiplicative, or A is discrete and ∥q∥ is an integer power of λ, or α = 0.

Proof. We can form a multiplicative number system Q = (A_{Q}, P_{Q}, ·, 1, ∥ ∥) generated by q by letting P_{Q} = {q}, A_{Q} = {q^{m} : m ≥ 0}, 1 = q^{0}, q^{m_{1}} ∗ q^{m_{2}} = q^{m_{1}+m_{2}}, and ∥q^{m}∥ = ∥q∥^{m}. Then A_{q} ≅ A ∗ Q, and Q has Property I, α_{Q} = 0. Now apply Theorem 68. □

If A is strictly multiplicative then Corollary 71 gives Theorem 9.69 from [6].

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Department of Pure Mathematics, University of Waterloo, Waterloo Ontario, N2L 3G1, Canada

kayeats@uwaterloo.ca

This paper is available via http://nyjm.albany.edu:8000/j/2002/8-4.html.