On the density of sumsets

Recently introduced by the authors in [Proc. Edinb. Math. Soc. 60 (2020), 139-167], quasi-densities form a large family of real-valued functions partially defined on the power set of the integers that serve as a unifying framework for the study of many known densities (including the asymptotic density, the Banach density, the logarithmic density, the analytic density, and the P\'olya density). We further contribute to this line of research by proving that (i) for each $n \in \mathbf N^+$ and $\alpha \in [0,1]$, there is $A \subseteq \mathbf{N}$ with $kA \in \text{dom}(\mu)$ and $\mu(kA) = \alpha k/n$ for every quasi-density $\mu$ and every $k=1,\ldots, n$, where $kA:=A+\cdots+A$ is the $k$-fold sumset of $A$ and $\text{dom}(\mu)$ denotes the domain of definition of $\mu$; (ii) for each $\alpha \in [0,1]$ and every non-empty finite $B\subseteq \mathbf{N}$, there is $A \subseteq \mathbf{N}$ with $A+B \in \mathrm{dom}(\mu)$ and $\mu(A+B)=\alpha$ for every quasi-density $\mu$; (iii) for each $\alpha \in [0,1]$, there exists $A\subseteq \mathbf{N}$ with $2A = \mathbf{N}$ such that $A \in \text{dom}(\mu)$ and $\mu(A) = \alpha$ for every quasi-density $\mu$. Proofs rely on the properties of a little known density first considered by R.C. Buck and the"structure"of the set of all quasi-densities; in particular, they are rather different than previously known proofs of special cases of the same results.


Introduction
Given X 1 , . . . , X n ⊆ Z, we denote by X 1 + · · · + X n the sumset of X 1 , . . . , X n (i.e., the set of all sums of the form x 1 + · · · + x n with x i ∈ X i for all i = 1, . . . , n); in particular, we write kX for the k-fold sumset (i.e., the sumset of k copies) of a given X ⊆ Z. Sumsets are some of the most fundamental objects studied in additive combinatorics [9,12], with a great variety of results relating the "size" of the summands X 1 , . . . , X n to that of the sumset X 1 + · · · + X n .
When the summands are finite, the size is usually the number of elements. Otherwise, many different notions of size have been considered, each corresponding to some real-valued function (either totally or partially defined on the power set of Z) that, while retaining essential features of a probability, is better suited than a measure to certain applications. In the latter case, the focus has definitely been on the asymptotic density d, the lower asymptotic density d ⋆ , and the Schnirelmann density σ, where we recall that, for a set X ⊆ N, with the understanding that the limit in the definition of d(X) has to exist. It is entirely beyond the scope of this manuscript to provide a survey of the relevant literature, so we limit ourselves to list a couple of classical results that are somehow related with our work: • In [14] (see, in particular, the last paragraph of the section "Added in proof"), B. Volkmann proved that, for all n ≥ 2 and α 1 , . . . , α n , β ∈ ]0, 1] with α 1 + · · · + α n ≤ β, there are A 1 , . . . , A n ⊆ N such that d(A i ) = α i for each i = 1, . . . , n and d(A 1 + · · · + A n ) = β. • In [10, Theorem 1], M. B. Nathanson showed that, for n ≥ 2 and all α 1 , . . . , α n , β ∈ [0, 1] with α 1 + · · · + α n ≤ β, there exist X 1 , . . . , X n ⊆ N with d ⋆ (X 1 ) = σ(X 1 ) = α i for each i = 1, . . . , n and d ⋆ (X 1 + · · · + X n ) = σ(X 1 + · · · + X n ) = β. Their proof combines the equidistribution theorem (i.e., that the sequence n → na mod 1 is uniformly distributed in the interval [0, 1] for every irrational number a) with the elementary property that, for every α ∈ ]0, 1], the asymptotic density of the set ⌊α −1 n⌋ : n ∈ N is equal to α. In the same manuscript, one can also find the following (see [3, Theorem 1.2]): This is a partial generalization of Theorem 1.1 for the special case where n = 1. A complete generalization was obtained by P.-Y. Bienvenu and F. Hennecart, shortly after [3] was posted on arXiv in Sept. 2018: Their proof is based on a "finite version" of Weyl's criterion for equidistribution due to P. Erdős and P. Turán (see [  In the present paper, we aim to prove that Theorems 1.1 and 1.2 and Proposition 1.3 hold, much more generally, with the asymptotic density d replaced by an arbitrary quasi-density µ (see Sect. 2.2 for definitions) and -what is perhaps more interesting -uniformly in the choice of µ (see Theorems 3.1-3.3 for a precise formulation). Most notably, this implies that Theorems 1.1 and 1.2 are true with d replaced by the Banach density [12,Sect. 5.7] or the analytic density [13, Sect. III.1.3], both of which play a rather important role in number theory and related fields and for which we are not aware of any similar results in the literature.
We emphasize that the proofs of our generalizations of Theorems 1.1 and 1.2 take a completely different route than the ones found in [1,3]: The latter critically depend on special features of the asymptotic density, whereas our approach relies on the properties of a little known density first considered by R. C. Buck [2] and the "structure" of the set of all quasi-densities. This is in line with one of our long-term goals, which was also the motivation for first introducing quasi-densities in [8]: Obtain sharper versions of various results in additive combinatorics and analytic number theory by shedding light on the "(minimal) structural properties" they depend on.

Preliminaries
In this section, we establish some notations and terminology used throughout the paper and prepare the ground for the proofs of our main theorems in Sect. 3.
2.1. Generalities. We denote by R the real numbers, by H either the integers Z or the non-negative integers N, and by N + the positive integers. Given x ∈ R, we use ⌊x⌋ for the greatest integer ≤ x and set frac(x) := x − ⌊x⌋; and given X ⊆ Z and h, k ∈ Z, we define k · X + h := {kx + h : x ∈ X}. An arithmetic progression of H is then a set of the form k · H + h with k ∈ N + and h ∈ H, and we write If X and Y are sets, then we write P(X) for the power set of X and X ⊆ fin Y to mean that X \ Y is finite. Further terminology and notations, if not explained when first introduced, are standard, should be clear from context, or are borrowed from [8].
2.2. Densities (and quasi-densities). We say a function µ ⋆ : P(H) → R is an upper density (on H) provided that, for all X, Y ∈ P(H), the following conditions are satisfied: k µ ⋆ (X) for every k ∈ N + and h ∈ H. In addition, we say µ ⋆ is an upper quasi-density (on H) if it satisfies (f1), (f3), and (f4).
It is arguable that non-monotone upper quasi-densities -whose existence is guaranteed by [8, Theorem 1] -are not so interesting from the point of view of applications. Yet, it seems meaningful to understand if monotonicity is "critical" to our conclusions or can be dispensed with: This is basically our motivation for considering upper quasi-densities in spite of our main interest lying in the study of upper densities (it is obvious that every upper density is an upper quasi-density).
With the above in mind, we let the conjugate of an upper quasi-density µ ⋆ be the function Then we refer to the restriction µ of µ ⋆ to the set as the quasi-density induced by µ ⋆ , or simply as a quasi-density (on H) if explicit reference to µ ⋆ is unnecessary. Accordingly, we call D the domain of µ and denote it by dom(µ).
Upper densities (and upper quasi-densities) were first introduced in [8] and further studied in [6,7]. Notable examples include the upper asymptotic, upper Banach, upper analytic, upper logarithmic, upper Pólya, and upper Buck densities, see [8,Sect. 6 and Examples 4, 5, 6, and 8] for details. In particular, we recall that the upper Buck density (on H) is the function where A is the collection of all finite unions of arithmetic progressions of H (as already mentioned in Sect. 2.1) and d ⋆ is the upper asymptotic density on N, that is, the function We shall write b ⋆ and b, respectively, for the conjugate of and the density induced by b ⋆ ; we call b ⋆ the lower Buck density and b the Buck density (on H). By [8, Note that the density induced by and the conjugate of d ⋆ are, resp., the asymptotic density d and the lower asymptotic density d ⋆ introduced in Sect. 1: One should keep this in mind when comparing our main results (that is, Theorems 3.1-3.3) with Theorems 1.1 and 1.2 and Proposition 1.3.

Basic properties.
Our primary goal in this section is to prove an inequality for the upper and the lower Buck density of sumsets of a certain special form (Proposition 2.4). We start with a recollection of basic facts that are either implicit to or already contained in [8].
Proposition 2.1. Let µ ⋆ be an upper quasi-density on H. The following hold: Proof. We have already mentioned that b ⋆ , as defined in Eq. (1), is an upper density and hence monotone. With this in mind, (i) follows from [8, Proposition 2(vi), Theorem 3, and Corollary 4], where among other things it is established that b ⋆ is the pointwise maximum of the set of all upper quasi-densities on H; (ii) follows from [8, Proposition 15] (which shows that b ⋆ is "shift-invariant") and the monotonicity of b ⋆ ; (iii) and (iv) follow from [8, Corollary 5 and Proposition 7]; and (v) follows from (i) and [8,Proposition 6].
The next result shows that b ⋆ and b ⋆ are additive under some circumstances. and On the other hand, we have by parts (iii) and (iv) of Proposition 2.1 that and it is a basic fact that, for all non-empty subsets S and T of R, So, putting it all together, we conclude from Eqs. (1) and (4) that in particular, it is seen from Eq. Proof. We can assume a = ∅, or else the conclusion is trivial. It is also clear that, if X = m · H + q for some m ∈ N + and finite q ⊆ H, then kX = m · H + kq ∈ A ; and it is obvious that k(A ∩ B) ⊆ kA ∩ kB (because x 1 + · · · + x k ∈ kA ∩ kB for all x 1 , . . . , x k ∈ A ∩ B). Since A ∩ B ∈ A and, by Proposition 2.1(iv), A ⊆ dom(b), we are thus left to check that (i) Pick x ∈ kA ∩ kB and, in case H = N, assume x ≥ k(k − 1)pq. Then x ≡ kr mod p and there exist a 1 , . . . , a k ∈ A with a 1 ≤ · · · ≤ a k such that x = a 1 + · · · + a k (observe that, if H = N, then a k ≥ (k − 1)pq). Since p and q are coprime, we gather from the Chinese remainder theorem that, for each In consequence, we find that a ′ 1 , . . . , a ′ k ∈ A ∩ B and hence x = a ′ 1 + · · · + a ′ k ∈ k(A ∩ B). This suffices to complete the proof, because x is an arbitrary element of (kA ∩ kB) \ V , where V := 0, k(k − 1)pq − 1 if H = N and V := ∅ otherwise (to wit, V is a finite set).
Proof. The "In particular" part of the statement is straightforward from Eq. (6) and Proposition 2.1(v), by the fact that mA ∈ A for all m ∈ N + and A ∈ A . So, we focus on the rest. Fix k ∈ 1, n , and define Since Y is a non-empty subset of q · H + t and p is coprime to q (by hypothesis), we gather from the Chinese remainder theorem that pqx + r ∈ Y ′ ⊆ V ′ = pq · H + r for some x ∈ H and r ∈ q · H + t. Using that X ′ is itself a finite union of arithmetic progressions modulo pq, it follows that, for all i ∈ N + and j ∈ N, and, on the other hand, in particular, the relation iX ′ + jr ⊆ fin iX ′ + j(pqx + r) becomes an equality when H = Z. Hence, and So taking into account that and considering that (k − i)t ≤ kt − (i + 1) + 1 ≤ nt < q for all i ∈ N (by hypothesis) and, by Eq. (8), we gather from Propositions 2.1(ii) and 2.2 and Eq. (9) We have Recalling that each of kS ′ , Z ′ k , and kV ′ is a finite union of arithmetic progressions (and hence, by Proposition 2.1(iv), a set in the domain of b) with kV ′ ⊆ q · H + kt (see Eq. (9)) and kt < q, it thus follows from Eq. (11), Lemma 2.3, and Propositions 2.1(v) and 2.2 that Moreover, we have from Eqs. (7) and (10) that Z ′ k ⊆ fin Z k ⊆ Z ′ k . Therefore, we conclude from the last display and Proposition 2.1(vi) that Z k ∈ dom(b) and b(Z k ) = b(Z ′ k ) = (pq) −1 kt (as wished).

A positional representation.
We introduce a non-standard positional representation of real numbers (Proposition 2.6) that will be of key importance in the proof of Theorem 3.1; cf. [11,Theorem 1.6] for an "analogous" result attributed by I. Niven to G. Cantor.
Proof. Since tα is irrational, the sequence (frac(N tα)) N ≥0 is equidistributed in Proposition 2.6. Let α be an irrational number in the interval [0, 1], and fix n ∈ N + . There then exist sequences (β i ) i≥1 and (q i ) i≥0 of positive integers with q 0 = 1 such that and, for every i ∈ N + , where we have defined in particular, β i is a positive integer because ⌊q i α i−1 ⌋ = n! k i for some k i ∈ N + (by definition of the set Q(α i−1 , nq 0 · · · q i−1 )), so that k i ≤ q i α i−1 /n! < k i + 1/n! and hence β i = k i . It is clear that On the other hand, α 0 = α ∈ ]0, 1[ ; and if α i−1 ∈ ]0, 1[ for some i ∈ N + , then it follows by Eqs. (13) and (15) that α i = q i α i−1 − n! β i ∈ ]0, 1[ . Thus, we see by induction that We may note, thanks to Eq. (14), that q i > q i α i−1 > n! ≥ 1, hence q i ≥ 2 for all i ∈ N + . To conclude, identity (12) is obtained by the fact that

Main results
This section is devoted to the main results of the paper. We start with a generalization of Theorem 1.1. Recall from Sect. 2.1 that A ∞ denotes the family of all subsets of H that can be expressed as the union of a finite set and countably many arithmetic progressions of H.
Proof. Thanks to Proposition 2.1(iii), it will be enough to prove that there exists A ∈ A ∞ such that kA ∈ dom(b) and b(kA) = αk/n for each k ∈ 1, n . To this end, we distinguish two cases.
Case 2: α is irrational. By Proposition 2.6, there exist sequences (β i ) i≥1 and (q i ) i≥0 of positive integers with q 0 = 1 such that gcd(q i , nq 0 · · · q i−1 ) = 1 for every i ∈ N + and Accordingly, we can recursively define sequences (X i ) i≥1 and (Y i ) i≥0 of subsets of H by taking Y 0 := H and, for each i ∈ N + , Because q 1 , q 2 , . . . are pairwise coprime integers, it is immediate from Eq. (17) and the Chinese remainder theorem that, for every i ∈ N + , there exists r i ∈ N such that Consequently, we obtain from Proposition 2.1(iv) that Note that the sets X 1 , X 2 , . . . are pairwise disjoint; moreover, Then, for each i ∈ N + , define A i := X 1 ∪ · · · ∪ X i and B i : It is obvious from Eq. (20) and our definitions that A ∈ A ∞ . So, to finish the proof, it only remains to show that kA ∈ dom(b) and b(kA) = kα/n for all k ∈ 1, n . For, fix k ∈ 1, n and i ∈ N + . Since b is monotone, it is clear from Eqs. (19) and (20) that On the other hand, it follows from Eq. (20) and the above that which in turn implies that We claim that For, let j ∈ 0, i − 1 and define Z i,j := A i \ A j = X j+1 ∪ · · · ∪ X i . We have from Eqs. (17) and (18) that In consequence, we see that Since each of X j+1 , Z i,j+1 , and Y j is a non-empty element of A , it thus follows from Proposition 2.4 This suffices to prove the claim (because X i = Z i,i−1 ), and in a similar way we find that The proof is essentially the same as the proof of Eq. (23), with the sets A i \ A j replaced by B i \ A j (0 ≤ j < i); we omit further details. Therefore, we gather from Eqs. (16), (21), (23), and (24) that Consequently, we see that  Proof. Similarly as in the proof of Theorem 3.1, it suffices to prove that there exists A ∈ A ∞ such that A + B ∈ dom(b) and b(A + B) = α. To this end, set x := min B and y := max B. We may assume without loss of generality that x = 0, because A + B = (A + x) + (B − x) and both A + x and B − x are subsets of H, with |B − x| = |B|. Therefore, B is a subset of N; and we can suppose that y = 0, or else the conclusion follows by Theorem 3.1. Now, the statement to be proved is trivial for α = 0 or α = 1 (just take A := ∅ in the former case and A := H in the latter). Consequently, let α ∈ ]0, 1[ and pick h, k ∈ N + such that h k < α < h + 1 k and h ≥ 2y + 1.
Then kα − h ∈ ]0, 1[ and h − y − 1 ≥ y, and we derive from Theorem 3.1 that there exists a set Then it is straightforward that and it follows by Propositions 2.1(iv) and 2.2 that Therefore, we find that On the other hand, Theorem 3.1 guarantees that b(Y ) = α for some Y ∈ dom(b). So, letting A := X ∪ Y and putting all pieces together, we get from Proposition 2.1(vi) that This finishes the proof, when considering that 0 ∈ Q ⊆ A and 1 ≤ gcd(A) ≤ gcd(Q) = 1.

Closing remarks
Looking at the statement of Theorem 3.1, it is natural to ask whether assuming A ∈ dom(µ), for some fixed quasi-density µ on H, is sufficient to guarantee that 2A ∈ dom(µ).
By [4,Proposition 2.2], the answer is negative for the asymptotic density d on N. But it follows by [8,Remark 3] that, in the classical framework of Zermelo-Fraenkel set theory with the axiom of choice, there is a density µ on H such that dom(µ) = H; hence, in this case, the answer is positive.
One can still wonder what happens with the Buck density b, especially in light of the role played by b in the proofs of Sect. 3. Again, the answer turns out to be in the negative. In fact, set V := {n! + n : n ∈ N} and A := {x 2 + y 2 : x, y ∈ V }.
Since b ⋆ is monotone, we gather from [6,Theorem 4.2], similarly as in the proof of Theorem 3.3, that A ∈ dom(b) and b(A) = 0. However, we will show that 2A / ∈ dom(b). To begin, we have Fix k ∈ N + and h ∈ N. By Lagrange's four square theorem, there exist y 1 , y 2 , y 3 , y 4 ∈ N such that h = y 2 1 + y 2 2 + y 2 3 + y 2 4 . Set, for each i ∈ 1, 4 , n i := (h + 1)k + y i and x i := n i ! + n i , and note that x i ∈ V , x i ≥ h, and n i ≥ k. It is then easily checked that Therefore