Bounding multiplicative energy by the sumset

By József Solymosi. 2009

180 citations recorded by the source corpus.

Abstract We prove that the sumset or the productset of any finite set of real numbers, A, is at least | A | 4 / 3 − e , improving earlier bounds. Our main tool is a new upper bound on the multiplicative energy, E ( A , A ) .

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/223b259202a15d4db0d72f47aca1c4a535f94ee2

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

1. Sophie Stevens
2. Ilya D. Shkredov
3. M. Rudnev
4. Oliver Roche-Newton
5. Brendan Murphy
6. Giorgis Petridis
7. E. Yazici
8. Frank de Zeeuw

In Discrete Mathematics and Combinatorics

1. Ilya D. Shkredov
2. Sophie Stevens
3. M. Rudnev
4. Brendan Murphy
5. Oliver Roche-Newton
6. E. Yazici
7. Giorgis Petridis
8. Frank de Zeeuw

In Geometry and Topology

1. M. Rudnev
2. Sophie Stevens
3. Oliver Roche-Newton
4. Brendan Murphy
5. Ilya D. Shkredov
6. Tom Sanders
7. Ben Joseph Green

Citations in the corpus

• On asymptotic formulae in some sum–product questions (2018, background): “At the moment the best results in the direction can be found in [52], [28], [29], [43] and in [1], [41] for R and Fp, respectively.”
• Growth in groups: ideas and perspectives (2013, unknown): “ver 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 theorem over R has a direct proof, also topological [Sol09].) Theorem 2.6 (Sum-product theorem [BKT04], [BGK06]; see also [EM03]). For any A⊂ F∗ p with ”
• NEW RESULTS ON SUM‐PRODUCT TYPE GROWTH OVER FIELDS (2017, background): “They were quite a way off the real case, where the main tools were the Szemerédi-Trotter theorem, see e.g. [9], and an order-based argument of Solymosi [51].”
• Growth Estimates in Positive Characteristic via Collisions (2015, background): “Both results are implicit in the paper of Solymosi and Tardos [32] and explicit in Jones’ paper [18].”
• New sum-product type estimates over finite fields (2014, background): “For the best results in the real and complex number cases, see respectively [23], [13].”
• 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.”