Množice števil brez tričlenih aritmetičnih zaporedij : magistrsko delo

Bone, Barbara (author), Jezernik, Urban (mentor)

V magistrskem delu predstavimo množice naravnih števil brez netrivialnih tričlenih aritmetičnih zaporedij, ki jih je motivirala domneva o prisotnosti k-členih aritmetičnih zaporedij v dovolj gostih množicah v naravnih številih. Najprej predstavimo algoritem iskanja množic brez treh števil, ki so enako narazen, in rezultate, ki smo jih dobili s poganjanjem programa. Spodnjo mejo za velikost množic brez tričlenih aritmetičnih zaporedij določimo z Behrendovo konstrukcijo. Za zgornjo mejo predstavimo najprej Meshulamov izrek, katerega dokaz je podoben dokazu Rothovega izreka. Oba izreka dokažemo s pomočjo diskretne Fourierove transformacije za primerne množice. V delu predstavimo tudi uporabo množic brez tričlenih aritmetičnih zaporedij za reševanje drugih problemov v matematiki. Zadnje poglavje posvetimo razreševanju prej omenjene domneve za štiri- in k-člena zaporedja v številih ter še aritmetičnim zaporedjem v praštevilih.

• Language: Slovenian
• Keywords: aritmetično zaporedje, Behrendova konstrukcija, diskrenta Fourierova transformacija, Meshulamov izrek, Rothov izrek
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2024
• PID: 20.500.12556/RUL-155586
• UDC: 511
• COBISS.SI-ID: 191763459
• Publication date in RUL: 07.04.2024

BONE, Barbara, 2024, Množice števil brez tričlenih aritmetičnih zaporedij : magistrsko delo [online]. Master’s thesis. [Accessed 14 April 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=155586

Secondary language

• Language: English
• Title: Sets of integers without three-term arithmetic progressions

Abstract

In this master's thesis, we present sets of natural numbers without three terms in an arithmetic progression, motivated by the conjecture about the presence of k-term arithmetic pogressions in sufficiently dense sets of natural numbers. First, we introduce an algorithm for finding examples of such sets and present the results obtained by running the program. We determine the lower bound for the size of sets without three-term arithmetic progressions using Behrend's construction. For the upper bound for the size of such sets, we first present Meshulam's theorem, whose proof is similar to the proof of Roth's theorem. Both theorems are proven using the discrete Fourier transformation for appropriate sets. The thesis also demonstrates the use of sets with no three terms in an arithmetic progression in solving other problems in mathematics. The final chapter is focused on resolving the mentioned conjecture for four-term and k-term arithmetic sequences in numbers and on primes that are equidistant from each other.

• Keywords: arithmetic progression, Behrend's construction, discrete Fourier transformation, Meshulam's Theorem, Roth's Theroem