On the Number of Sums and Products

By József Solymosi. 2005

184 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/5099dbf85ce7e3a4fa0b1666e22f6d378323c842

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Discrete Mathematics and Combinatorics

Citations in the corpus

Growth in groups: ideas and perspectives (2013, unknown): “known for long (Erd˝os-Szemer´edi). The version over ﬁnite ﬁelds is harder, since there is no natural topology or fully natural ordering to work with. (Over R, there is a brief and very natural proof [Ele97] based on a result that is essentially topological [ST83]; the best bound for the sum-product”
THE SUM-PRODUCT ESTIMATE FOR LARGE SUBSETS OF PRIME FIELDS (2007, background): “Explicit versions of (1) have been obtained in [5]–[8].”
On the energy variant of the sum-product conjecture (2016, methodology): “This improved the previous best-known exponent 43 obtained some ten years earlier as a result of a graceful and renowned construction by Solymosi [28], which only relies on the order properties of reals and does not use the Szemerédi-Trotter theorem.”
On the size of the set A(A + 1) (2008, background): “2Key words and phrases. sums, products and expanding maps.”
Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order (2006, background): “) Estimate (2) was improved in the series of papers [11, 13, 20].”
An improved point‐line incidence bound over arbitrary fields (2016, methodology): “The first bound was originally obtained by Roche-Newton, Rudnev and Shkredov [33], but here we recover it using Elekes’ original argument for (2).”
On the size of k-fold sum and product sets of integers (2003, methodology): “Solymosi [So] using the Szemerédi-Trotter theorem in incidence geometry.”
An Explicit Sum-Product Estimate in p (2010, background): “In the corresponding problem for integers (i.e., if the field Fp is replaced by the set of integers) the conjecture of Erd¨os and Szemer´edi [6] is that max{|A + A|, |AA|} ≫ c(")|A|2 " for any " > 0. At present the best known bound in the integer problem is max{|A + A|, |AA|} ≫ |A|14/11(log |A|) 3/1”
NEW RESULTS ON SUM‐PRODUCT TYPE GROWTH OVER FIELDS (2017, methodology): “ided that M|A|3 ≤ p2. In this section, we will improve the ﬁrst of the statements, in the form of Theorem 4. The method of proof of Theorem 4 is as follows. Firstly, we modify an argument of Solymosi [51], by using Theorem 10 in place of the Szemer´edi-Trotter Theorem. This leads to an upper bound o”
New sum-product type estimates over finite fields (2014, methodology): “The same trick works for applications of Szemerédi-Trotter type bounds for sum-product estimates over the real and complex fields, enabling the improvements in [20] and [15] over the foundational result in [5].”
Open problems in additive combinatorics (2007, unknown): “We tried to include primarily those problems we came across in our research, or at least lying close to the area of our research interests.”