5 research outputs found
Note on the Theorem of Balog, Szemer\'edi, and Gowers
We prove that every additive set with energy has a
subset of size such that
. This is, essentially, the largest
structured set one can get in the Balog-Szemer\'edi-Gowers theorem
Products of Differences over Arbitrary Finite Fields
There exists an absolute constant such that for all and all
subsets of the finite field with elements, if
, then Any suffices for sufficiently large
. This improves the condition , due to Bennett, Hart,
Iosevich, Pakianathan, and Rudnev, that is typical for such questions.
Our proof is based on a qualitatively optimal characterisation of sets for which the number of solutions to the equation is nearly
maximum.
A key ingredient is determining exact algebraic structure of sets for
which is nearly minimum, which refines a result of Bourgain and
Glibichuk using work of Gill, Helfgott, and Tao.
We also prove a stronger statement for when are sets in a prime field,
generalising a result of Roche-Newton, Rudnev, Shkredov, and the authors.Comment: 42 page
Entropy methods for sumset inequalities
In this thesis we present several analogies betweeen sumset inequalities and entropy inequalities. We offer an overview of the different results and techniques that have been developed during the last ten years, starting with a seminal paper by Ruzsa, and also studied by authors such as Bollobás, Madiman, or Tao. After an introduction to the tools from sumset theory and entropy theory, we present and prove many sumset inequalities and their entropy analogues, with a particular emphasis on Plünnecke-type results. Functional submodularity is used to prove many of these, as well as an analogue of the Balog-Szemerédi-Gowers theorem. Partition-determined functions are used to obtain many sumset inequalities analogous to some new entropic results. Their use is generalized to other contexts, such as that of projections or polynomial compound sets. Furthermore, we present a generalization of a tool introduced by Ruzsa by extending it to a much more general setting than that of sumsets. We show how it can be used to obtain many entropy inequalities in a direct and unified way, and we extend its use to more general compound sets. Finally, we show how this device may help in finding new expanders