Sums, products and ratios along the edges of a graph

In their seminal paper Erd\H{o}s and Szemer\'edi formulated conjectures on the size of sumset and product set of integers. The strongest form of their conjecture is about sums and products along the edges of a graph. In this paper we show that this strong form of the Erd\H{o}s-Szemer\'edi conjecture does not hold. We give upper and lower bounds on the cardinalities of sumsets, product sets and ratio sets along the edges of graphs.

for some δ > 0.Here and in what follows we use the asymptotic notation Ω(•), O(•) and Θ(•).For two functions over the reals, f (x) and g(x), we write f (x) = Ω(g(x)) if there is a positive constant, B > 0, and a threshold, D, such that f (x) ≥ B • g(x) for all x ≥ D. We write f (x) = O(g(x)) if there is a positive constant, B > 0, and a threshold, D, such that f (x) ≤ B • g(x) for all x ≥ D. Finally, f (x) = Θ(g(x)) if f (x) = O(g(x)) and f (x) = Ω(g(x)).
Erdős and Szemerédi formulated an even stronger conjecture.In this variant one considers a subset of the possible pairs in the sumset and product set.Let G n be a graph on n vertices, v 1 , v 2 , . . ., v n , with n 1+c edges for some real c > 0. Let A be an n-element set of real numbers, A = {a 1 , a 2 , . . ., a n }.The sumset of A along G n , denoted by A + Gn A, is the set {a i + a j |(i, j) ∈ E(G n )}.The product set along G n is defined similarly, The Strong Erdős-Szemerédi Conjecture is the following.Conjecture 2. [5] For every c > and ε > 0, there is a threshold, n 0 , such that if n ≥ n 0 then for any n-element subset of integers A ⊂ N and any graph G n with n vertices and at least n 1+c edges The original conjecture, inequality (1), would follow from this stronger conjecture by taking the complete graph, G n = K n .
For more details on the sum-product problem, we refer to a recent survey [6].
Here we refute Conjecture 2 by giving constructions with small sumsets along a graph where the product set is also small.A similar problem -which is closely related to the original sum-product conjecture -is to bound the number of sums and ratios along the edges of a graph.We give upper and lower bounds on these quantities.

Products
In the next construction we define a set and a graph with many edges such that both the sumset and the product set are small.Theorem 3.For arbitrary large m 0 , there is a set of integers, A, and graph on |A| = m ≥ m 0 vertices, G m , with Ω(m 5/3 / log 1/3 m) edges such that Proof: It is easier to describe our construction using rational numbers instead of integers.Multiplying then with the least common multiple of the denominators will not effect the size of the sumset or the product set, giving a construction for integers.
We define the set A first and then the graph.Below, the function lpf(m) denotes the least prime factor of m.We write (v, w) = 1 if v and w are relatively prime.
The products of pairs of elements of A along an edge of G m are integers of size at most n 4/3 .The sums along the edges are of the form The denominator is a positive integer of size at most n 1/3 and the numerator is a positive integer of size at most 2n, hence the number of sums is at most 2n 4/3 .
Modifying the construction above we give a counterexample to the Strong Erdős-Szemerédi Conjecture for every 1 > c > 0. For the sake of simplicity we will ignore logarithmic multipliers, using the asymptotic notations Ω l (•), O l (•) and Θ l (•).For two functions over the reals, f (x) and g(x), we write f (x) = Ω l (g(x)) if there is a constant, B ≤ 0, and a threshold, D, such that f Theorem 4. For every 1 > c > 0 there is a δ > 0, such that for arbitrary large n there is an n-element subset of integers, A ⊂ N, and a graph H n with Ω l (n 1+c ) edges such that Proof: We consider two cases separately, when 0 < c ≤ 2/3 and when 2/3 < c < 1.

Case 1.
(2/3 < c < 1) We define A similar to the previous construction, but now the ranges of u, v and w are different.
The number of u, w, v triples satisfying the conditions is Ω l (n).If |A| = m then let H m be the graphs with vertex set A. Two elements, a, b ∈ A are connected by an edge if they can be written as a = uw v and b = vz w .There are at least Ω l (n 1+c ) edges in H m .The products of two such elements of A are integers of size at most n 2c .A typical sum is The numerator is an integer of size at most 2n and the denominator is an integer of size at most n 1−c .Therefore the sumset along the edges of H m has size at most 2n 3) It is possible to describe a construction similar to that in the first case, but we prefer to take a subgraph of the graph G m in Theorem 3. Let p be a parameter satisfying 0 < p ≤ 1, to be specified later.In G m take first the edges with the pm 4/3 most popular products.This gives a graph G m with at least Ω l (pm 5/3 ) edges and with O l (pm 4/3 ) products and at most O l (m 4/3 ) sums.Now in G m take the most popular pm 4/3 sums to get the subgraph H m with at least Ω l (p 2 m 5/3 ) edges, with O l (pm 4/3 ) products and O l (pm 4/3 ) sums.Choosing p to be n c/2 /n 1/3 we get Ω l (n 1+c ) edges and O l (n 1+c/2 ) sums and products.

Ratios
In this section, we consider a problem similar to the Strong Erdős-Szemerédi Conjecture, but we change products to ratios.Define (Note that each edge (i, j) here providse two ratios: a i /a j and a j /a i .) Changing product to ratio is a common technique in sum-product bounds.When one is using the multiplicative energy (like in [10] and [7] for example) then the role of product and ratio are interchangeable.The multiplicative energy of a set A is the number of quadruples (a, b, c, d) ∈ A 4 such that ab = cd, which is clearly the same as the number of quadruples where a/c = d/b.But the symmetry fails in the Strong Erdős-Szemerédi Conjecture.We are going to show examples when the sumsets and ratiosets are even smaller than in the previous construction.
3.1.Connection to the original conjecture.What is the connection of the Strong Erdős-Szemerédi Conjecture to the original conjecture (when G n = K n )?Similar questions were investigated in [2].Here we consider the connections to the sum-ratio problem along a graph.If there was a counterexample to Conjecture 1, that would imply the existence of a set with very small sumset and ratioset along a dense graph.In our first result let us suppose that both the product set and ratioset are small.A bound on the cardinality of the product set does not imply a similar bound on the ratio set.Or, equivalently, a bound on the cardinality of the sumset does not imply a similar bound on the difference set.A classical construction of the second author [9]  Note that in this construction, the multiplicity of a member a − a ′ = k−1 i=0 (α(a) − α(a ′ ))10 i of A − A along the edges of the complete graph on A is 3 r , where r is the number of indices satisfying α(a) = α(a ′ ).It is easy to see that for every fixed small δ > 0 the fraction of edges in which the parameter r exceeds (1/3 + δ)k is at most e −Ω(δ 2 k) .Therefore, any graph on A with at least (9 k ) 1−cδ 2 edges, for an appropriate absolute positive constant c, has on at least half of its edges a value of the difference with multiplicity at most 3 (1/3+δ)k , implying that the number of distinct differences along the edges is at least 0.5 (9 k ) 1−cδ 2 3 (1/3+δ)k .As 9/3 1/3 > 6 and the number of sums is only 6 k , this shows that for small δ the number of differences along the edges of any such graph is significantly larger than the number of sums.
The above discussion shows that we need a modified statement to transform a possible counterexample to Conjecture 1 to a statement about few sums and ratios along a graph.We are going to apply the following lemma.Lemma 6.Let A be an n-element subset of an abelian group and suppose that |A + A| ≤ K|A|.Then there is an integer parameter, M, and a graph, H n , with vertex set A, and at least With these notations we can write the additive energy as There is a k such that (3) The number of the edges is ti∈T k m(t i ).From inequality (3) we have a lower bound on m(t i )-s, so the number of edges is at least In order to bound the magnitude of M, note that since ti∈T k m(t i ) ≤ |A| 2 , the largest m(t i ) for an element t i in T k satisfies the trivial inequality max ti∈T k (m(t i )) ≤ 2|A| 2 /M.Replacing m(t i )-s by 2|A| 2 /M on the left side of inequality (3) we get the desired upper bound on M. The lower bound follows from the same inequality and from the fact that m(t i ) ≤ |A| for every t i ∈ A − A.
In the proof of Theorem 5, in G N , the edges were defined by pairs of vertices having the form (a − ζac, b + ζac).If we have a bound on the product set only, |AA| ≤ n 2−β , then in our new graph, G ′ N , we connect a − ζac and b + ζac only if (a, b) is an edge in H n , where H n is defined in Lemma 6 applied to the set A in the multiplicative group.This guarantees that the ratio set along the edges is not (much) larger than the product set along edges of G N .The new graph G ′ N is a subgraph of the graph G N in Theorem 5.
The new parameter, M, makes the description of our next result a bit complicated, but the important feature of this construction is that it shows that if there is a counterexample to Conjecture 1 then there is a set of numbers, B, and graph, G ′ N , with many edges so that the sumset and the ratio set are both small.The number of edges might be less than in Theorem 5, but then the size of the ratio set along this graph is much smaller.We will state a simpler, but weaker statement in a corollary below.
Theorem 7. Let us suppose that there is a set of n real numbers, A, such that |A + A| ≤ n 2−α and |AA| = Θ l (n 2−β ) for some α > 0, β > 1/2 real numbers.Then there is a set B with N > n elements, a parameter M in the range and Note that the number of edges in G ′ N is at least Ω l (N 2+β 3−β ) which is bigger than the cardinalities of the sumset and ratio set along its edges.
Corollary 8. Let us suppose that there is a set of n real numbers, A, such that |A + A| ≤ n 2−α and |AA| = Θ l (n 2−β ) for some α > 0, β > 1/2 real numbers.Then there is a set B with N > n elements, and a graph G ′ N with Ω l (N Proof of Theorem 7: Applying Lemma 6 to the multiplicative subgroup of real numbers with K = |A| 1−β we get a graph H n as in the lemma.We connect a − ζac and b + ζac in G ′ N only if (a, b) is an edge in H n .For given a and b we can choose any c ∈ A, so the number of edges is Ω l (|A| M |A| 2+β ) = Ω l ( M |A| 4+β ).The size of the ratio set along the edges is |A|M, and the sumset is not larger than O(N 3.2.Constructions.In the next construction we define a set and a graph with many edges such that both the sumset and the ratio set are very small.It can be viewed as a special case of Theorem 5 with α = 0 and β = 1. Theorem 9.For arbitrary large n, there is a set of reals and graph, G n with Ω(n 3/2 ) edges such that Proof: As before, in the construction we define the set A first and then Two elements, 2 i − 2 j and −(2 k − 2 ℓ ), are connected by an edge iff j = ℓ.Along this G n both the sumsets and the ratiosets are small, and the number of edges is a little more than 3.3.Matchings.Erdős and Szemerédi mentioned in their paper that maybe even for a linear number of edges (when c = 0) the Strong Erdős-Szemerédi Conjecture holds, but noted that it is not true for reals.There are sets of reals such that G n is a perfect matching and It was shown by Alon, Angel, Benjamini, and Lubetzky in [1] that if we assume the Bombieri-Lang conjecture (see details in [2]), then for any set of integers A, if G n is a matching, It is possible that 4/7 can be improved to a number close to 1, but if we change multiplication to ratio, then just the trivial bound, Ω( √ n), holds.A simple construction demonstrating this is the following.Take n = k 2 and distinct primes, p 1 , ., p k , q 1 , .., q k .The matching consists of all pairs (p i /q j , (q j − 1)p i /q j ) (i, j = 1, .., k).Then all these rationals are pairwise distinct, the sums along the matching edges are the p i s, the quotients (of large divided by small) along the edges are (q j − 1).For this set A and matching G n we have |A + Gn A| + |A/ Gn A| = O(|A| 1/2 ), which is as small as possible.

Lower bounds
Lower bounds on the number of sums and products along graphs were obtained in [1].Under assuming the Bombieri-Lang conjecture they proved that if A is an n-element set of integers and G n a graph with m edges then n 28/9+o (1) .
We will apply a variant of Elekes' proof, used in his sum-product estimate in [4] to get better estimates.
The four pencils are The vertical lines with a point in P, The horizontal lines with a point in P, L 2 := {y = −2 i + 2 j |1 ≤ j ≤ i ≤ √ n}.
The slope −1 lines with a point in P, Lines through the origin with a point in P, Note that in the definition of L 3 and L 4 the same lines are listed multiple times.Ignoring these repetitions it is easy to see that all four families have size approximately n, and |P | = Ω(n 3/2 ).One can apply a projective transformation to shift the centers of the pencils from infinity to R 2 .

Theorem 5 .
Let us suppose that there is a set of n real numbers, A, such that |A + A| ≤ n 2−α ,|AA| = Θ(n 2−β ) and |A/A| ≤ n 2−β for some α, β > 0 real numbers.Then there is a set B with N > n elements and a graph G N with Ω(N 3 3−β ) edges such that |B/ GN B| = O(N ), and |B + GN B| = O(N 2−α 3−β ).Proof: Let B = {a ± ζbc | a, b, c ∈ A}, where ζ ∈ R is selected such that all sums are distinct, B has cardinality 2|A||AA| = N = Θ(n 3−β ).In the graph, G N , every a−ζac is connected to b+ζac by an edge.The number of edges is n 3 = Ω(N 3 3−β ).The number of sums along the edges is |A + A| = O(N 2−α 3−β ), and ratios along the edges have the form a + ζac b − ζac = 1 + ζc b/a − ζc so the cardinality of the ratioset is at most |A||A/A| = O(N ).
is an example for that.It uses the observation that S = {0, 1, 3} satisfies |S + S| = 6 and |S − S| = 7.If we consider the set of numbers, A, of the form a = k−1 i=0 α i (a)10 i , where α i (a) ∈ S, then |A| = 3 k , |A + A| = 6 k , and |A − A| = 7 k .

M|A| 3 4K
log |A| edges, such that |A − Hn A| ≤ M.Moreover, M satisfies the following inequalities: |A| K log |A| ≤ M ≤ 4K log |A||A|.Proof: The additive energy of A, denoted by E(A), is the number of a, b, c, d quadruples from A such that a + b = c + d.This is the same as the number of quadruples satisfying a − c = d − b.By the Cauchy-Schwartz inequality E(A) ≥ |A| 3 /K.Denote the elements of the difference set as follows A − A = {t 1 , t 2 , . . ., t ℓ }.For every element we can define its multiplicity, m( ) edges such that