Bachelor's Thesis - Finite Difference Sets

Supervisor: doc. RNDr. Tatiana Jajcayova, PhD.

Student: Erik Szalay

Aim: The main topic of this project is a part of the combinatorial number theory. We will study generalized difference sets on subsets of positive integers whose differences include each positive integer with a prescribed number of repetitions. The twofold aim of the project is to approach the classification of admissible repetition sequences of finite generalized difference sets and the design of algorithms that produce generalized difference sets with additional properties for specified admissible sequences. The part of the project is to program and analyse developed algorithms.

Thesis Timeline

30.10.2018: Thesis web page
13.11.2018: Thesis timeline
13.11.2018: Application for testing properties of sequences, testing of small sequences
27.11.2018: Finalization of collecting and studying sources
12.1.2019: Writing up theoretical context and background and designing prototype of software tools
31.1.2019: Testing of medium sequences, optimization of approximation
31.3.2019: Testing of large sequences, optimization of approximation, sequence generation
30.4.2019: First version of Thesis
15.5.2019: Finalization of Thesis

Download

First Chapter of the Bachelor’s Thesis and Prototype

Sources

Publications:

Tatiana B. Jajcayová, Robert Jajcay: Notes on subtractive properties of natural numbers, 2008

Tatiana B. Jajcayová: Generalized Difference Sets

Swara Kopparty: Results on Frequency Sequences

J. H. van Lint, R. M. Wilson: A Course in Combinatorics, 2nd edition
Cambridge University Press 2001
ISBN-13 978-0-511-67289-7

Emily H. Moore , Harriet S. Pollatsek: Difference Sets: Connecting Algebra, Combinatorics, and Geometry
American Mathematical Society, 2013
ISBN-13: 978-0821891766

Other Theses:

Marek Štefaňák: Frequency Sequences of Finite Difference Sets
Bachelor’s Thesis, 2014
Study programme: Applied Computer Science
Field of study: 9.2.9 Applied Informatics
Department: Department of Applied Informatics
Supervisor: RNDr. Tatiana Jajcayová

Ivana Kellyérová: Generalized difference sets
Bachelor’s Thesis, 2014
Study programme: Applied Computer Science
Field of study: 2511 Applied Informatics
Department: Department of Applied Informatics
Supervisor: RNDr. Tatiana Jajcayová

Bc. Marián Opial: Generalized difference sets – An Algorithmic Approach
Master’s Thesis, 2017
Study programme: Applied Computer Science
Field of study: 2511 Applied Informatics
Department: Department of Applied Informatics
Supervisor: RNDr. Tatiana Jajcayová

Presentation

Progress

3/4/2019 - Included a condition that removed 33% of wrong sequences.
3/18/2019 - Wrote a chapter about a different way to think about Generalized Difference Sets; that there exists an isomorphism between the sets and complete undirected weighted graphs, which satisfy a specific condition.
4/2/2019 - Wrote a chapter describing the conditions that eliminate wrong sequences that I discovered with proofs.