|
|
Roth's theorem on arithmetic progressions
Krkavec, Michal ; Klazar, Martin (advisor) ; Kráľ, Daniel (referee)
Title: Roth's theorem on arithmetic progressions Author: Michal Krkavec Department: Department of Applied Mathematics Supervisor: doc. RNDr. Martin Klazar, Dr., Department of Applied Mathematics Abstract: In the presented summary work we study sets of natural numbers not containing arithmetic progressions. The aim of this thesis is to provide an overview and comparison of both analytical and combinatorial proofs of Roth's theorem, which states that every set of positive upper asymptotic density contains arithme- tic progression of length three. We also focus on the Erd˝os-Turán conjecture and Szemerédi's theorem, which finally settled the conjecture for arithmetic progres- sions of arbitrary length k. In the end, we introduce the bounds for the number r3(n), which corresponds to the largest size of a subset A ⊆ [n], which contains no arithmetic progressions of length three. At the end we present two constructions of progression-free sets. Keywords: Additive number theory, Arithmetic progressions, Roth's theorem, Elkin's construction 1
|
|
|
Roth's theorem on arithmetic progressions
Krkavec, Michal ; Klazar, Martin (advisor) ; Kráľ, Daniel (referee)
Title: Roth's theorem on arithmetic progressions Author: Michal Krkavec Department: Department of Applied Mathematics Supervisor: doc. RNDr. Martin Klazar, Dr., Department of Applied Mathematics Abstract: In the presented summary work we study sets of natural numbers not containing arithmetic progressions. The aim of this thesis is to provide an overview and comparison of both analytical and combinatorial proofs of Roth's theorem, which states that every set of positive upper asymptotic density contains arithme- tic progression of length three. We also focus on the Erd˝os-Turán conjecture and Szemerédi's theorem, which finally settled the conjecture for arithmetic progres- sions of arbitrary length k. In the end, we introduce the bounds for the number r3(n), which corresponds to the largest size of a subset A ⊆ [n], which contains no arithmetic progressions of length three. At the end we present two constructions of progression-free sets. Keywords: Additive number theory, Arithmetic progressions, Roth's theorem, Elkin's construction 1
|
|
|
Sequences and series (not only) in word problems
FIŘTOVÁ, Petra
The dissertation is comprised of list of tasks with the sequences and set of numbers used to the high schools. This issue is very extensive and the dissertation is focused on application of the sequesces and set of numberss in word mathematic tasks which are thematicly devided into particular tasks according to their specialization. The tasks are classi?ed from the point of historical view as well as from the point of application view - in planimetry and stereometry, in ?nancial mathematics etc. In the part dedicated to sequences is stated extra chapter focused on excersises from mathematic olympics for high schools.
|
|
| |
guest ::
