Probabilistic regularity lemma and its applications in combinatorics

Sa？etak This paper begins with two well-known theorems from additive combinatorics, which are then reduced to a result from graph theory. A further generalization is formulated in the language of probability theory, in order to be established using the probabilistic variant of the Szemerédi regularity lemma. This lemma provides a decomposition of an arbitrary random variable into a structured part, a pseudorandom part, and an error, and the paper presents its complete proof