Director: András Szűcs

I am extremely grateful to my supervisor, Márton Elekes for calling my attention to many inspiring problems, for useful discussions and his continuous support. He taught me almost everything I know about research, and his kind personality made the joint work pleasant. I would like to express my greatest thanks to Miklós Laczkovich for helping me not only in mathematics but also in real-world problems. I am deeply indebted to my co-author, Zoltán Buczolich for the inspiring problem and fruitful collaboration. I would like to thank Miklós Laczkovich for his lectures and for the Real Analysis Problem Solving Seminar, organized by Márton Elekes, Tamás Keleti and Tamás Mátrai. These lectures had great influence on me, without them I would have probably never began my doctoral studies. I am grateful to András Máthe ́ for his valuable suggestions. His deep insight and interest in the whole mathematics had a great impact on me. I am indebted to Tibor Tómács for his support in LATEX. I would like to thank my colleagues both in Budapest and Eger for the friendly atmosphere. I am grateful to my high school math teacher, Jánosne ́ Tóth for showing me for the first time how beautiful mathematics can be. Finally, I would like to thank my parents for their continuous support. Content