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.
|
