ON THE DENSITY OF SUMSETS, II

Abstract Arithmetic quasidensities are a large family of real-valued set functions partially defined on the power set of 
$\mathbb {N}$
 , including the asymptotic density, the Banach density and the analytic density. Let 
$B \subseteq \mathbb {N}$
 be a nonempty set covering 
$o(n!)$
 residue classes modulo 
$n!$
 as 
$n\to \infty $
 (for example, the primes or the perfect powers). We show that, for each 
$\alpha \in [0,1]$
 , there is a set 
$A\subseteq \mathbb {N}$
 such that, for every arithmetic quasidensity 
$\mu $
 , both A and the sumset 
$A+B$
 are in the domain of 
$\mu $
 and, in addition, 
$\mu (A + B) = \alpha $
 . The proof relies on the properties of a little known density first considered by Buck [‘The measure theoretic approach to density’, Amer. J. Math. 68 (1946), 560–580].


Introduction
Let d be the asymptotic (or natural) density on the non-negative integers N and dom(d) be the family of all sets X ⊆ N which possess asymptotic density, meaning that the limit of 1 n X ∩ [1, n] as n → ∞ exists.We are going to show that, if B ⊆ N is non-empty and "sufficiently small", then there is a family of sets of the form A + B with A and A + B both in dom(d) such that the corresponding asymptotic densities attain every value in the interval [0, 1], where A + B := {x + y : x ∈ A, y ∈ B} is the sumset of A and B (see Sect. 2 for details and examples).Writing P := {2, 3, 5, . ..} for the set of primes, we obtain as a special case the following: Theorem 1.1.For each α ∈ [0, 1], there exists A ∈ dom(d) such that A + P ∈ dom(d) and d(A + P) = α.
In fact, our main result (Theorem 2.3) is much more general and stronger.Not only does it allow us to show that Theorem 1.1 holds with the primes replaced by a greater variety of sets and the asymptotic density d replaced by any in a large class of axiomatically defined "densities" µ (including, among others, the Banach density, the analytic density, and the logarithmic density), but also it does so uniformly in the choice of µ.The result, whose proof relies on the structural properties of a little known density first considered by Buck [2], belongs to a vast literature on the interplay between the structure of sumsets and their "largeness", see, e.g., [1,6,7,9,13,15,17] and [3,8,14,16].
An analogue of Theorem 1.1 with P replaced by a non-empty finite set B ⊆ N (and without the additional requirement that A ∈ dom(d)) was proved by Faisant et al. in [6,Theorem 2.2].A previous attempt to extend the latter to an infinite set B was made by Chu in [4,Theorem 1.5].However, it turned out that the proof is flawed [5].Therefore, Theorem 1.1 provides the first example of an infinite set satisfying the required claim.(The proof is based on completely different ideas from [4,6].)Notation.We use Z for the integers and N + for the positive integers.Given X ⊆ N and q ∈ N, we define q • X := {qx : x ∈ X} and X + q := X + {q}.We let an arithmetic progression (AP) be a set of the form k • N + h with k ∈ N + and h ∈ N, and we denote by A the family of all finite unions of APs.Finally, for each a, b ∈ Z we write a, b := [a, b] ∩ Z for the discrete interval between a and b.

Preliminaries and Main Result
We say that a real-valued function µ ⋆ defined on the power set P(N) of N is an arithmetic upper density (on N) if, for all X, Y ⊆ N, the following conditions are satisfied: for every k ∈ N + and h ∈ N.Moreover, we call µ ⋆ an arithmetic upper quasi-density (on N) if it satisfies (f1), (f3), and (f4).
Remark 2.1.While there do exist non-monotone arithmetic upper quasi-densities [11, Theorem 1], such functions are not so interesting from the point of view of applications.Nevertheless, it seems meaningful to understand if monotonicity is critical to certain conclusions or can instead be dispensed with.This is our motivation for considering arithmetic upper quasi-densities in spite of our main interest lying in the study of arithmetic upper densities (of course, the latter are a special case of the former).
We let the conjugate of an arithmetic upper quasi-density µ ⋆ be the function µ ⋆ : P(N) → R : X → 1 − µ ⋆ (N \ X), and we refer to the restriction µ of µ ⋆ to the set as the arithmetic quasi-density induced by µ ⋆ , or simply as an arithmetic quasi-density (on N) if explicit reference to µ ⋆ is unnecessary.Accordingly, we call D the domain of µ and denote it by dom(µ).
Arithmetic upper [quasi-]densities and arithmetic [quasi-]densities were introduced in [11] and further studied in [10,12,13], though we are adding here the adjective "arithmetic" to emphasize that they assign precise values to APs (see Proposition 3.1(iv) below).
Notable examples of arithmetic upper densities include the upper asymptotic, upper Banach, upper analytic, upper logarithmic, upper Pólya, and upper Buck densities, see [11,Sect. 6 and Examples 4, 5, 6, and 8] for details.In particular, we recall that the upper Buck density (on N) is the function where d ⋆ is the upper asymptotic density (on N), i.e., the function We will write b ⋆ and b, respectively, for the conjugate of and the density induced by b ⋆ .
Remark 2.2.The asymptotic density d in Theorem 1.1 is nothing but the density induced by d ⋆ .
We are ready to state the main theorem of the paper, whose proof we postpone to Sect. 3.
The sets B ⊆ N such that b(B) = 0 have been studied in [12], where they are called "small sets".Since b is monotone and subadditive, it is clear that the family of small sets is closed under finite unions and subsets.Examples of small sets include the finite sets, the factorials, the perfect powers, and the primes.One may be tempted to conjecture that a set B ⊆ N is small if it is "sufficiently sparse".However, the property of being small depends on the distribution of B through the APs of N (see Proposition 3.1(v)).E.g., the set {n! + n : n ∈ N} is not small (its upper Buck density is 1), but is sparse by any standard.
To date, it is not known whether non-monotone arithmetic quasi-densities do exist (cf.Remark 2.1).However, arithmetic quasi-densities satisfy a weak form of monotonicity (implicit in the proof of Proposition 3.1) that will be enough for our goals.

Proofs
To start with, we collect some basic properties of [upper and lower] quasi-densities that will be used, possibly without further comment, in the proof of Theorem 2.3.Proposition 3.1.Let µ ⋆ be the conjugate of an arithmetic upper quasi-density µ ⋆ on N, and µ be the density induced by µ ⋆ .Then the following hold: Proof.See [13, Proposition 2.1] for items (i)-(iv) and [12,Proposition 2.6] for item (v).As for (vi), let X ∈ A and Y ⊆ N.There then exist k ∈ N + and H ⊆ 0, k − 1 such that X = k • N + H, and hence with the understanding that an empty union is the empty set).
We are ready for the proof of our main result.Note that the special case of a non-empty finite B ⊆ N was settled in [13, Theorem 1.2] by a different argument.
Proof of Theorem 2.3.If α = 1, then the conclusion is obvious (by taking A = N).So, we assume from now on that 0 ≤ α < 1.We divide the remainder of the proof into a series of three claims.
Claim A. There exist a sequence (H n ) n≥1 of (non-empty) subsets of N with H n ⊆ 0, n! − 1 and a sequence (h n ) n≥1 with h n ∈ H n such that, for all n ≥ 1, the following hold (we set H ′ n := H n \ {h n } for ease of notation): Proof of Claim A. (i) We proceed by induction.The base case is clear by taking H 1 := {0} and h 1 := 0.
As for the inductive step, fix m ≥ 1 and suppose we have already found a set H m ⊆ 0, m! − 1 and an integer h m ∈ H m such that the claimed inequality holds for n = m.Since Consequently, we define It is thus clear from Eq. ( 4) and the minimality of k m+1 that the claimed inequality is also true for n = m + 1.By induction, this is enough to complete the proof.
(ii) This is now a straightforward consequence of the recursive construction of the sequences (H n ) n≥1 and (h n ) n≥1 as given in the inductive step of the proof of item (i).

Claim B. Set
On the other hand, Claim A(ii) gives A n+1 ⊆ A n .Since b is monotone, it follows that b(A n ) tends to a limit as n → ∞.So, we get from Eq. ( 6 Proof.Fix n ∈ N + .We gather from Claim A(ii) that Considering that, by Proposition 3.1 Now, fix ε > 0. Since b(B) = 0 there exists n ε ∈ N + such that B covers at most ε • n! residue classes modulo n! for all n ≥ n ε .Consequently, we obtain from the subadditivity of b ⋆ and Claim A(i) that On the other hand, the first inequality in the last display and Claim A(i) imply Therefore, we get from Eqs.As a final remark, we point out that, mutatis mutandis, all the results of this paper carry over to arithmetic [upper] quasi-densities on Z, in the same spirit of [10,11,12,13].
(7)-(9) that α − ε < b ⋆ (A + B) ≤ b ⋆ (A + B) ≤ α + ε for every ε > 0, which suffices to conclude that A + B ∈ dom(b) and b(A + B) = α.By Proposition 3.1(iii) and Claims B and C, this is enough to finish the proof of the theorem.Proof of Theorem 1.1.Straightforward from Theorem 2.3 an Remark 2.2, when considering that the Buck density of the set of primes is zero (as already noted in Sect.2).