On the density of sumsets and product sets
Norbert Hegyv\'ari, Fran\c{c}ois Hennecart, P\'eter P\'al Pach

TL;DR
This paper explores the relationships between the density of integer sets and the densities of their sumsets, product sets, and subset sums, revealing new insights into their combinatorial structure.
Contribution
It introduces novel links between the densities of sets and their sumsets, product sets, and subset sums, advancing understanding in additive and multiplicative combinatorics.
Findings
Established bounds relating set density to sumset density
Identified conditions under which product sets have increased density
Connected the density of subset sums to original set properties
Abstract
In this paper some links between the density of a set of integers and the density of its sumset, product set and set of subset sums are presented.
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Taxonomy
TopicsAnalytic Number Theory Research · Limits and Structures in Graph Theory · Graph theory and applications
On the density of sumsets and product sets
Norbert Hegyvári
Norbert Hegyvári, ELTE TTK, Eötvös University, Institute of Mathematics, H-1117 Pázmány st. 1/c, Budapest, Hungary
,
François Hennecart
François Hennecart, Univ. Jean-Monnet, Institut Camille Jordan CNRS 5208, 23 rue Michelon, 42023 Saint-Étienne cedex 2, France
and
Péter Pál Pach 1
Péter Pál Pach, Budapest University of Technology and Economics, 1117 Budapest, Magyar tudósok körútja 2, Hungary
Abstract.
In this paper some links between the density of a set of integers and the density of its sumset, product set and set of subset sums are presented.
1 Supported by the National Research, Development and Innovation Office of Hungary (Grant Nr. NKFIH (OTKA) PD115978 and NKFIH (OTKA) K124171) and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.
1. Introduction and notations
In the field of additive combinatorics a popular topic is to compare the densities of different sets (of, say, positive integers). The well-known theorem of Kneser gives a description of the sets having lower density such that the density of is less than (see for instance [9]). The analogous question with the product set is apparently more complicated.
For any set of natural numbers, we define the lower asymptotic density , the upper asymptotic density and the asymptotic density in the natural way:
[TABLE]
known as the lower and upper asymptotic densities. If the two values coincide, then we denote by the common value and call it the asymptotic density of .
Throughout the paper denotes the set of positive integers and . We will use the notion for and . For functions we write (or ), if there exists some such that for large enough .
In Section 2 we investigate the connection between the (upper-, lower-, and asymptotic) density of a set of integers and the density of its sumset. In Section 3 we give a partial answer to a question of Erdős by giving a necessary condition for the existence of the asymptotic density of the set of subset sums of a given set of integers. Finally, in Section 4 we consider analogous problems for product sets.
2. Density of sumsets
For subsets of an additive monoid , the sumset is defined to be the set of all sums with , . For the following clearly hold:
[TABLE]
We shall assume that our sets are normalized in the sense that contains [math] and .
First observe that there exists a set of integers not having an asymptotic density such that its sumset has a density: for instance has lower density , upper density and its sumset has density , since it contains every nonnegative integer. For this kind of sets , we denote respectively
[TABLE]
and we have
[TABLE]
The first question arising from this is to decide whether or not for any such that there exists a set of integers such that . This question has no positive answer in general, though the following weaker statement holds.
Proposition 2.1**.**
Let . There exists a normalized set such that and .
Proof.
Let be a thin additive basis, that is, a basis containing 0 and satisfying as . Now, let . Then is a normalized set satisfying and .
(Note that is also an appropriate choice for .) ∎
Remark**.**
We shall mention that Faisant et al [1] proved the following related result: for any and any positive integer , there exists a sequence such that , .
After a conjecture stated by Pichorides, the related question about the characterisation of the two-dimensional domains has been solved (see [3] and [6]).
Note that if the density exists, then , and have to satisfy some strong conditions. For instance, by Kneser’s theorem, we know that if for some set we have , then is, except possibly a finite number of elements, a union of arithmetic progressions in with the same difference. This implies that must be a rational number. From the same theorem of Kneser, we also deduce that if , then is an arithmetic progression from some point onward. It means that is a unit fraction, hence contains any sufficiently large integer, if we assume that is normalized.
Another strong connection between and can be deduced from Freiman’s theorem on the addition of sets (cf. [2]). Namely, every normalized set satisfies
[TABLE]
A related but more surprising statement is the following:
Proposition 2.2**.**
There is a set of positive integers for which does exist and does not exist.
Proof.
Let us take and , then observe that
[TABLE]
Let be a sufficiently quickly increasing sequence of integers with , , and define by
[TABLE]
Then has density . Moreover, for any
[TABLE]
thus , if we assume .
We also have
[TABLE]
hence using again the assumption that . ∎
For any set having a density, let
[TABLE]
then we have
[TABLE]
A question similar to the one asked for can be stated as follows: given such that , does there exist a set such that ?
We further mention an interesting question of Ruzsa: does there exist and a constant such that for any set having a density,
[TABLE]
Ruzsa proved (unpublished) that in case of an affirmative answer, we necessarily have .
3. Density of subset sums
Let be a sequence of positive integers. Denote the set of all subset sums of by
[TABLE]
Zannier conjectured and Ruzsa proved that the condition implies that the density exists (see [8]). Ruzsa also asked the following questions:
- i)
Is it true that for every pair of real numbers , there exists a sequence of integers for which ? This question was answered positively in [5].
- ii)
Is it true that the condition also implies that exists ?
We shall prove the following statement.
Proposition 3.1**.**
Let be a sequence of positive integers. Assume that for some function satisfying we have
[TABLE]
*where .
Then exists.*
Proof.
We first prove that there exists a real number such that
[TABLE]
Let be large enough. Then
[TABLE]
Since , we have , thus
[TABLE]
On the other hand,
[TABLE]
since . Therefore,
[TABLE]
Observe that , hence letting
[TABLE]
we obtain from (1) and (2) that
[TABLE]
Now, we show that . Since
[TABLE]
the condition implies that from (4) we obtain that . Therefore, in fact, for large enough we have with some . Now, let be a fixed integer. For we have
[TABLE]
since . Hence, indeed holds.
Therefore, using the assumption on we obtain that . So (3) yields that
[TABLE]
Therefore, the sequence has a limit which we denote by . Furthermore, observe that
[TABLE]
The next step is to consider an arbitrary sufficiently large positive integer and decompose it as
[TABLE]
where and are defined in the following way. (Here is a fixed, sufficiently large positive integer.) The index is chosen in such a way that . If , then , otherwise is the largest index for which . The indices are defined similarly. We stop at the point when the next index would be at most and set . As , we have
[TABLE]
Furthermore, let
[TABLE]
(The empty sum is , as usual.)
Let and for let and consider
[TABLE]
Note that in this union each element appears at most once, since according to the definition of the sets are pairwise disjoint as
[TABLE]
holds for every .
The set of those elements of that are not covered by is:
[TABLE]
Therefore,
[TABLE]
Using and (6), we obtain that , where (as ). (Note that .)
That is, the set covers with the exception of a “small” portion of size . Therefore, by letting the density of the uncovered part tends to 0.
Let us consider . If a sum is contained in , then the sum of the elements with indices larger than is . Otherwise, the sum is either at most or at least .
Therefore, .
Hence,
[TABLE]
Therefore,
[TABLE]
and
[TABLE]
Now, observe that
- •
by (6),
- •
by using and ,
- •
by using (5). Letting this term is also of size .
Hence, we obtain from (7) and (8) that .
∎
4. Density of product sets
For any semigroup and any subset , we denote by the product set
[TABLE]
In this section we focus on the case , the semigroup (for multiplication) of all positive integers. The restricted case is even more interesting, since implies .
The sets of integers satisfying the small doubling hypothesis are well described through Kneser’s theorem. The similar question for the product set does not plainly lead to a strong description. We can restrict our attention to sets such that , since by setting we have and .
Examples**.**
i) Let be the set of all non-squarefree integers. Letting we have and
[TABLE]
ii) However, while , we have
[TABLE]
iii) Furthermore, the set of all squarefree integers satisfies
[TABLE]
since consists of all cubefree integers.
iv) Given a positive integer , the set A_{k}=\big{\{}n\in\mathbb{N}\,:\,\gcd(n,k)=1\big{\}} satisfies
[TABLE]
where is Euler’s totient function.
We have the following result:
Proposition 4.1**.**
For any positive there exists a set such that and .
Proof.
Assume first that .
For let , then . Therefore, and . If , then satisfies the requested condition. Since , an appropriate can be chosen for every
Assume now that .
Let be the increasing sequence of prime numbers and
[TABLE]
The complement of the set contains exactly those positive integers that are not divisible by any of , thus we have
[TABLE]
Similarly, the complement of the set contains exactly those positive integers that are not divisible by any of or can be obtained by multiplying such a number by one of . Hence, we obtain that
[TABLE]
Note that
[TABLE]
As , moreover and are increasing sequences satisfying (9) and , we obtain that is covered by . That is, for every we have for some , and then is an appropriate choice.
∎
We pose two questions about the densities of and .
Question**.**
If and , then , too. Given two integers , the set
[TABLE]
is multiplicatively stable. What are the sets of positive integers such that or less restrictively
[TABLE]
Question**.**
It is clear that implies , since .
For any we denote
[TABLE]
Is it true that for any or at least for ?
The next result shows that the product set of a set having density 1 and satisfying a technical condition must also have density 1.
Proposition 4.2**.**
Let be a set of positive integers with asymptotic density . Furthermore, assume that contains an infinite subset of mutually coprime integers such that
[TABLE]
Then the product set also has density .
Proof.
Let be arbitrary and choose a large enough such that
[TABLE]
Let be a large integer. For any , the set contains all the products with . We shall use a sieve argument. Let be a finite subset of and for some . For any , let
[TABLE]
Observe that
[TABLE]
Then
[TABLE]
By the inclusion-exclusion principle we obtain
[TABLE]
whence
[TABLE]
where the empty intersection denotes the full set .
For any finite set of integers we denote by the least common multiple of the elements of . Now, we consider
[TABLE]
By the assumption we immediately get
[TABLE]
Plugging this into (11):
[TABLE]
Since the elements of are mutually coprime,
[TABLE]
(Note that for the empty product is defined to be 1, as usual.) Since we get
[TABLE]
by our assumption (10). Thus finally
[TABLE]
This ends the proof. ∎
Remark**.**
Specially, the preceding result applies when contains a sequence of prime numbers such that . For this it is enough to assume that
[TABLE]
However, we do not know how to avoid the assumption on the mutually coprime integers having infinite reciprocal sum. We thus pose the following question:
Question**.**
Is it true that implies ?
An example for a set such that and .
According to the fact that the multiplicative properties of the elements play an important role, one can build a set whose elements are characterized by their number of prime factors. Let
[TABLE]
where denotes the number of prime factors (with multiplicity) of . An appropriate generalisation of the Hardy-Ramanujan theorem (cf. [4] and [10]) shows that the normal order of is and the Erdős-Kac theorem asserts that
[TABLE]
which implies . Now we prove that . The principal feature in the definition of is that must contain almost all integers such that .
For let
[TABLE]
Let us consider first the density of the integers such that
[TABLE]
Let be a large number and write
[TABLE]
By a theorem of Hildebrand (cf. [7]) on the estimation of , the number of -friable integers up to , we conclude that the above cardinality is . Hence, we may avoid the integers satisfying (12). By the same estimation we may also avoid those integers for which .
Let be an integer such that and
[TABLE]
Our goal is to find a decomposition with , .
Let
[TABLE]
where . We also assume that . Let . Then
[TABLE]
Let
[TABLE]
where . Then , which yields
[TABLE]
On the other hand,
[TABLE]
Now , hence
[TABLE]
and
[TABLE]
Therefore, the following statement is obtained:
Proposition 4.3**.**
The set
[TABLE]
has density [math] and its product set has density .
By a different approach we may extend the above result as follows.
Theorem 4.4**.**
For every there exists a set such that , and .
Proof.
We start with defining a set such that and . Let us choose a subset of the primes such that . Such a subset can be chosen, since . Now, let denote the -th prime and let
[TABLE]
[TABLE]
Furthermore, let
[TABLE]
and
[TABLE]
Let . Clearly, contains exactly those numbers that do not have any prime factor in , so . For and the probability that an integer does not have any prime factor being less than from is . Therefore, , and consequently also holds. If , then satisfies the conditions. From now on let us assume that .
Our aim is to define a subset in such a way that and . As we will have and . The set is defined recursively. We will define an increasing sequence of integers and sets () satisfying the following conditions (and further conditions to be specified later):
- (i)
,
- (ii)
,
- (iii)
.
That is, is obtained from by dropping out some elements of in the range . Finally, we set .
Let and . We define the sets in such a way that the following condition holds for every with some depending only on :
[TABLE]
Since , a threshold can be chosen in such a way that holds for with this choice of . Now, assume that and are already defined for some . We continue in the following way depending on the parity of :
- Case I:
is odd.
Let be the smallest integer such that
[TABLE]
for some . We claim that such an exists, indeed it is at most . For we have
[TABLE]
Hence, is well-defined (and ). Let and . (Specially, it can happen that and .) Note that satisfies .
- Case II:
is even.
Now, let be the smallest index for which .
We have and is obtained from by deleting finitely many elements of it: , where . As , we have that
[TABLE]
therefore, . So for some we have that , that is, is well-defined. Let and . Clearly, satisfies .
This way an increasing sequence and sets are defined, these satisfy conditions (i)-(iii). Finally, let us set . Note that .
We have already seen that implies that and . At first we show that . Let be arbitrary. If is large enough, then . As satisfies and we obtain that
[TABLE]
This holds for every , therefore, .
As a next step, we show that . Let be odd. According to the definition of and there exists some such that
[TABLE]
For brevity, let . As we get that . Also,
[TABLE]
since . Thus . Clearly , and as we have , therefore .
Finally, we prove that . Let be even. According to the definition of and , we have . However, , therefore , thus as it was claimed.
∎
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 2[2] G.A. Freiman, Foundations of a structural theory of set addition (translated from the Russian), Translations of Mathematical Monographs, Vol 37. American Mathematical Society, Providence, R. I., 1973.
- 3[3] G. Grekos, D. Volkmann, On densities and gaps , J. Number Theory 26 (1987), 129–148.
- 4[4] G.H. Hardy, S. Ramanujan, The normal number of prime factors of a number , Quart. J. Math. 48 (1917), 76–92.
- 5[5] N. Hegyvári, Note on a problem of Ruzsa , Acta Arith. 69 (1995), 113–119.
- 6[6] F. Hennecart, On the regularity of density sets , Tatra Mt. Math. Publ. 31 (2005), 113–121.
- 7[7] A. Hildebrand, On the number of positive integers ≤ x absent 𝑥 \leq x and free of prime factors > y absent 𝑦 >y , J. Number Theory 22 (1986), 265–290.
- 8[8] I.Z. Ruzsa, The density of the set of sums , Acta Arith. 58 (1991), 169–172.
