Difference sets are not multiplicatively closed

We prove that for any finite set A of real numbers its difference set D:=A-A has large product set and quotient set, namely, |DD|, |D/D| \gg |D|^{1+c}, where c>0 is an absolute constant. A similar result takes place in the prime field F_p for sufficiently small D. It gives, in particular, that multiplicative subgroups of size less than p^{4/5-\eps} cannot be represented in the form A-A for any A from F_p.

Problem. Let A, P ⊂ R be two finite sets, P ⊆ A − A. Suppose that |PP| ≤ |P| 1+ε , where ε > 0 is a small parameter. In other words, P has small product set. Is it true that there exists δ = δ (ε) > 0 with ∑ x∈P |{a 1 − a 2 = x : a 1 , a 2 ∈ A}| |A| 2−δ ? (1.1) Thus, we consider a set P with small product set and we want to say something nontrivial about the additive structure of A, that is, the sum of additive convolutions over P. The question plays an important role in recent papers [15], [22] and [28], [29]. Even the famous unit distance problem of Erdős (see [4] and also the survey [20]) can be considered as a question of that type. Indeed, the unit distance problem is to find a good upper bound for where A is a finite subset of the Euclidean plane (which we consider as the complex plane) and S 1 is the unit circle. Since the unit circle S 1 is a subgroup, it trivially has small product set: S 1 · S 1 = S 1 .
Let us return to (1.1). In [15] Roche-Newton and Zhelezov studied some sum-product type questions and obtained the following result (in principle, the method of their paper allows one to obtain subexponential bounds for the sum from (1.1) but not of the required form). The multiplicative energy of A, denoted by E × (A), is the number of solutions to the equation ab = cd, where a, b, c, d ∈ A.
Theorem 1 For any ε > 0 there are constants C (ε),C (ε) > 0 such that for any set A ⊂ C one has E × (A − A) ≤ max{C (ε)|A| 3+ε , |A − A| 3 exp(−C (ε) log 1/3−o(1) |A|)} . (1.2) Thus, bound (1.2) says us that the difference set D = A − A enjoys a non-trivial upper bound for its multiplicative energy. The proof used a deep result of Sanders [16] and the Subspace Theorem of Schmidt (see, e.g., [20]). Roughly speaking, thanks to the Subspace Theorem, Roche-Newton and Zhelezov obtained an upper bound for the sums from (1.1) for P equal to a multiplicative subgroup of C of small rank (so P automatically has small product set for trivial reasons) and using Sanders' structural result they extended it to general sets with small multiplicative doubling -see details in [15]. Theorem 1 has the following consequence.
Corollary 2 Let A ⊂ R be a finite set, let D = A − A, and let ε > 0 be a real number. Then for some constant C (ε) > 0 one has |DD|, |D/D| ε |D| · min{|D| 3 |A| −(3+ε) , exp(C (ε) log 1/3−o(1) |D|)} .  The bound (1.4) can be considered as a new necessary condition for a set to be a difference set of the form A − A. Namely, any such set must have a large product set and a large quotient set.
One might think that the optimal version of (1.4) should state that |DD|, |D/D| |D| 2−ε for arbitrary ε > 0, but this is not true: see Proposition 22 which gives examples of sets A with |DD|, |D/D| |D| 3/2 .
Also, it was conjectured in [15] The authors obtained some first results in this direction. A refined version of Theorem 3, Theorem 20 below, implies the following result.
A simple consequence of the conjectured bound (1.1) is that A − A = P for sets P with small product/quotient set. Our weaker estimate (1.4) gives the same, so, in particular, geometric progressions are not (symmetric) difference sets of the form A − A. An analog of geometric progressions in prime fields F p are multiplicative subgroups. Obtaining an appropriate version of Theorem 3 in the finite-fields setting and using further tools, we obtain the following theorem.
The results of the article allow us to make a first tiny tiny step towards answering a beautiful question of P. Hegarty [6].
Problem. Let P ⊆ A + A be a strictly convex (concave) set. Is it necessarily true that |P| = o(|A| 2 )?
Recall that a sequence of real numbers A = {a 1 < a 2 < · · · < a n } is called strictly convex (concave) if the consecutive differences a i − a i−1 are strictly increasing (decreasing). It is known that from a combinatorial point of view sets P with small product set have behaviour similar to convex (concave) sets (see [19] or discussion before Corollary 36), but in the opinion of the author they have a simpler structure. The following proposition is a consequence of Theorem 3.
Thus, Corollary 6 proves the conjecture of Hegarty in the case of pure difference sets P instead of sumsets and where we assume that P has small product/quotient set instead of assuming convexity.
In the proof we develop some ideas from [22], combining them with the Szemerédi-Trotter Theorem (see section 3), as well as with a new simple combinatorial observation, see formula (4.3). The last formula tells us that if one forms a set D/D, D := A − A (which is known as Q[A] in the literature, see e.g. [27]), then the set D/D contains a large subset R ⊆ D/D which is additively rich. Namely, we consider and note that R = 1 − R. By the Szemerédi-Trotter Theorem the existence of such additive structure in R means that the product of R is large and hence the product of D is large as well. The paper is organized as follows. In section 2 we give a list of the results, which will be further used in the text. In the next section we discuss some consequences of the Szemerédi-Trotter Theorem in its uniform and modern form. In Section 4 we prove our main Theorem 20 which implies Theorem 3 and Corollary 4. In the next section we deal with the prime fields case and obtain Theorem 5 above. Finally, the constants in Theorem 3 and Corollary 4 can be improved in the case of the quotient set D/D but it requires much more work -see section 6. In the appendix we discuss some generalizations of the quantities from section 3.
Let us conclude with a few comments regarding the notation used in this paper. All logarithms are to base 2. The signs and are the usual Vinogradov symbols. When the constants in the signs depend on some parameter M, we write M and M .
The author is grateful to D. Zhelezov and S. Konyagin for useful discussions, and to O. Roche-Newton and M. Rudnev who pointed out to him how to improve Theorems 20, 24. Also, he thanks the anonymous referees for a careful reading of the text and for helpful suggestions.
Lemma 8 Let A, B,C ⊆ G be three finite sets. Then In this paper we have to deal with the quantity T(A, B,C, D), see [14], [22] (T standing for collinear triples) In [7] Jones proved a good upper estimate for the quantity T(A), A ⊂ R (another proof was obtained by Roche-Newton in [14]). Upper bounds for T(A) when A belongs to a prime field can be found in [1] and [22]. (2.5) Theorem 10 Let p be a prime number and let A ⊂ F p be a subset with |A| < p 2/3 . Then The method of our paper relies on the famous Szemerédi-Trotter Theorem [26], see also [27]. Let us recall the relevant definitions.
We call a set L of continuous plane curves a pseudo-line system if any two members of L have at most one point in common. Define the number of incidences I(P, L) between points and pseudo-lines to be I(P, L) = |{(p, l) ∈ P × L : p ∈ l}|.
Theorem 11 Let P be a set of points and let L be a pseudo-line system. Then

Remark 12
If we redefine a pseudo-line system as a family of continuous plane curves with O(1) points in common and if any two points are simultaneously incident to at most O(1) curves, then Theorem 11 remains true: see e.g. [27,Theorem 8.10] for precise bounds in this direction. Now let us recall the main result of [23].
Theorem 13 Let Γ ⊆ F p be a multiplicative subgroup, let k ≥ 1 be a positive integer, and let x 1 , . . . , x k be different nonzero elements of F p . Also, let The same holds if one replaces Γ in (2.6) by any cosets of Γ.
Thus, the theorem above asserts that |Γ where α k , β k are some sequences of positive numbers, and α k , β k → 0 as k → ∞.
We finish this section with a result from [22], see Lemma 19.

Lemma 14
Let Γ ⊆ F p be a multiplicative subgroup, and k be a positive integer. Then for any nonzero distinct elements x 1 , . . . , x k of F p one has where |θ | ≤ 1.

Some consequences of the Szemerédi-Trotter Theorem
In this section we discuss some implications of Szemerédi-Trotter Theorem 11, which are given in a modern form (see e.g. [13]). We start with a definition from [21].
holds for every finite set B ⊂ R and every real number τ ≥ 1.
So, D(A) can be considered as the infimum of numbers such that (3.1) takes place for any B and τ ≥ 1 but, of course, the definition is applicable just for sets A with small quantity D(A). Now we can introduce a new characteristic of a set A ⊂ R, which can be considered as a generalization of D(A).
holds for every finite set B ⊂ R and every real number τ ≥ 1.
The usual quantity D(A) corresponds to the case Φ(x, y) = x + y.
Any SzT-type set has small number of solutions of a wide class of linear equations, see, e.g., [10, Corollary 8] (where nevertheless another quantity D(A) was used) and [21,Lemmas 7,8], say. This is another illustration of that fact in a more general context. Then

3)
and as required. Estimate (3.4) can be obtained similarly, or see the proof of Lemma 7 in [21]. Also notice that this bound, combined with the Cauchy-Schwarz inequality, implies (3.3). This concludes the proof. 2 In [12] authors considered the quantitỹ where the infimum is taken over convex/concave functions f and proved that D + (A) d (A). In a similar way, one can consider the quantity D × (A), which corresponds to Φ(x, y) = xy, and obtain where the infimum is taken over all functions f such that for any The last bound is a generalization of [8, Lemma 15]. For the rigorous proof of formula (3.6) and the proof of a similar upper bound for the quantity D Φ (A), see the appendix.

The proof of the main result
First of all let us derive a simple consequence of Theorem 11.
Lemma 18 Let A, B,C, D ⊂ R be four finite sets. Then for any nonzero α one has In a dual way  Similarly, in order to prove (4.2), we consider the equation ab = α, a ∈ A, b ∈ B, and also the equation (p − c)(p * − d) = α, p ∈ A +C, p * ∈ B + D, which correspond to the curves l p,d = {(x, y) : (p − x)(y − d) = α} and the points P = C × (B + D). It is easy to check that any two curves l p,d , l p ,d have at most two points in common. After that one applies the Szemerédi-Trotter Theorem one more time and performs the calculations above. This completes the proof. 2 Take arbitrary finite sets A, B ⊂ R, |B| > 1 and consider the quantity    Combining the last two bounds, we obtain the first inequality from (4.5). The second inequality follows from Theorem 19. The trivial estimates |A| ≤ |D| ≤ |A| 2 imply formula (4.6). This completes the proof. 2 From identity (4.3) and the results of section 3, it follows that the set R has some interesting properties, which we give in the next proposition, one consequence of which is that R has a large product set |RR| |R|  One can obtain the inequality |RR| |R| 5 4 by taking B = R in the general formula (4.8). Actually, since Proposition 21 gives the stronger result D × (R) |RR| 2 /|R| 2 , one can apply methods of [21] to derive the inequality |RR| |R| 14/9 · D × (R) −5/9 and hence the inequality |RR| |R| 24/19 .
We finish this section with an example of a set A with a difference set that has small product set and small quotient set.

P r o o f. Let
and again |D/D| ≤ 25|D| 3/2 . This completes the proof. 2

The finite fields case
In this section p is a prime number. An analog of Lemma 18 in the prime fields setting is the following, see [1].
Lemma 23 Let A, B ⊂ F p be two subsets with |A| = |B| < p 2/3 . Then for any nonzero α ∈ F p one has |A ∩ (B + α)| |A| −1/2 |AB| 4/3 . Combining the last two bounds, we obtain the first inequality from (5.5  Hence, |R| |D| 5/6 ≥ p 5/9 , which is a contradiction. Again, the second inequality in (5.5) follows from Theorem 10 and the fact that |A| ≤ |R| p 5/9 < p 2/3 . This completes the proof. 2 Note that an analog of Proposition 21 also holds in F p because of an appropriate version of Theorem 11 (see [1]). Also note that one can relax the condition |R[A]| ≤ cp 5/9 at the cost of a weaker bound (5.5).
Theorem 26 Let A ⊂ R be finite and let X ⊆ (A − A) \ {0} be a set with X = −X. Then Using the Cauchy-Schwarz inequality, we get Applying Theorem 9, we obtain the result. This completes the proof. 2 Note that we do not use the fact that A is a subset of the reals to get formula (6.3), but just that A belongs to some field. Now we obtain the main result of this section. We have σ D (A) ≥ |A| 2 /2. Now let where j ≥ 1. By the pigeonhole principle there exists j L such that σ D j (A) |A| 2 /L . Considering the set X ∪ (−X), we see that the last bound holds for this new set with possibly bigger constants. Redefining X to be X ∪ (−X) we conclude that the bound holds for a symmetric set. After that, using the arguments of the proof of Theorem 20, identity (6.1), and Lemma 18 with Substituting the bound (6.6) as well as estimate (6.2) of Theorem 26, we obtain We have σ X (A) ≥ ∆|X|2 −1 and hence In view of inequality (6.5) and the bound |X| ≥ |D |/2, we know that Substituting the last estimate into (6.8) and recalling that ∆ |A| 2 /|D|, we obtain, finally, The same arguments, combined with the method of proof of Theorem 24, give us the following theorem. P r o o f. With the notation of the proof of Theorem 27, we obtain an analog of inequality (6.7), namely, Of course one needs to apply Theorem 10 instead of Theorem 9 in the proof. Using the last formula we obtain after some calculations the result in the case |D| < p 2/3 . For larger sets we suppose that (6.9) does not hold and obtain that |D| ≤ |DD|, |D/D| |D| The last inequality contradicts the assumption that |A| p 10/27 log 4/9 |A|. This completes the proof. 2

Appendix
We finish our paper by studying the quantity D Φ (A) and obtaining the bound (3.6).
Definition 29 For any function Φ : R 2 → R and any finite set A ⊂ R, let where the infimum is taken over all functions F such that Lemma 32 Let A ⊂ R be a finite set and let Φ : R 2 → R be a function such that the quantity d Φ (A) is defined. Then P r o o f. By the assumption, for any C in the infimum from (7.1) we have that |F(A,C)| ≥ max{|A|, |C|}. Thus, |F(A,C)| 2 |A||C| ≥ 1, and hence d Φ (A) ≥ 1. Taking C equal to a one-element set and applying a trivial bound |F(A,C)| ≤ |A|, we obtain that d Φ (A) ≤ |A|.
Let us prove the second part of the lemma. Take a set C and a function F such that where ε > 0 is arbitrary, such that for any element a ∈ A we have that |F({a},C)| ≥ |C| and that for allÃ ⊆ A the inequality |F(Ã,C)| ≥ |Ã| holds. Clearly, for an arbitrary a ∈ A we have again |F({a},C)| ≥ |C| and for allÃ ⊆ A we have the inequality |F(Ã,C)| ≥ |Ã|. Using the trivial inequality |F(A ,C)| ≤ |F(A,C)|, we obtain If A = {0}, F(x, y) = xy, and Φ(x, y) = xy, say, then it is easy to see that d Φ (A) = 0. Thus, we need the property |F({a},C)| ≥ |C| for every a ∈ A to obtain d Φ (A) ≥ 1. Now we can prove the main technical statement of this section.
Proposition 33 Let A, B ⊂ R be finite sets and let Φ : R 2 → R be any function such that for any fixed z and a ∈ A we have the inequality |{b ∈ B : Φ(b, z) = a}| ≤ M, where M > 0 is an absolute constant. Then for every real number τ ≥ 1 one has In other words, A has SzT Φ -type with O M (d Φ (A)).
P r o o f. First of all note that we can assume that τ ≤ |B|, since otherwise the left-hand side of inequality (7.3) is equal to 0. Further, by hypothesis, for any fixed z and any a ∈ A one has |{b ∈ B : Φ(b, z) = a}| ≤ M and hence τ ≤ min{|B|, M|A|} . To derive the last inequality we have used estimate (7.4), the bound |F(A,C)| ≥ |A| and the trivial inequality τ 2 ≤ (min{|B|, M|A|}) 2 ≤ |B| · M|F(A,C)| .
This concludes the proof. 2 Example 34 Let F(x, y) = x + y and let Φ(x, y) = y sin x. Then the lines l b,s = {(x, y) : s = y + x sin b} form a pseudo-line system. Nevertheless, for any fixed z one has |Φ −1 (·, z)| = +∞ or zero. So, the pseudo-line condition does not imply any bounds for the size of preimages of the map Φ.
Remark 35 It is easy to see that it is enough to check the condition |{b : Φ(b, z) = a}| ≤ M, a ∈ A just for z belonging to the set from the left-hand side of (7.3).
Now let us derive some consequences of Proposition 33. Inequality (7.6) in the corollary below was obtained in [18]: see Lemma 2.6. Here we give a more systematic proof. The corollary asserts that D + (A) < 4 for any convex/concave set A. On the other hand, from Proposition 33 it follows that D + (A) ≤ M 2 for any A with |AA| ≤ M|A| or |A/A| ≤ M|A|. This demonstrates some combinatorial similarity between convex sets and sets with small product/quotient set.
Corollary 36 Let A ⊂ R be a finite convex set. Then D + (A) < 4. Furthermore, if A ⊆ A is a subset of A, then for an arbitrary set B ⊂ R one has |A + B| |A | 3/2 |B| 1/2 |A| −1/2 . These curves have infinite cardinality and form a pseudo-line system. Clearly, for any nonempty A, C such that 0 / ∈ A,C one has |F({a},C)| = |C| for every a ∈ A, and |F(Ã,C)| ≥ |Ã| holds for an arbitrarỹ A ⊆ A.