ABSTRACT
A common problem in additive number theory is to prove that a sequence with certain properties exists. One of the successful ways to obtain an affirmative answer for such a problem is to use the probabilistic method, established by Erdős. To show that a sequence with a property https://www.w3.org/1998/Math/MathML"> P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780138747022/abb99e87-ffb7-4196-a4c4-acb0033e3d6a/content/eq3679.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> exists, it suffices to show that a properly defined random sequence satisfies https://www.w3.org/1998/Math/MathML"> P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780138747022/abb99e87-ffb7-4196-a4c4-acb0033e3d6a/content/eq3680.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> with positive probability. The value of the probabilistic method has been demonstrated by the fact that in most problems solved by it, it seems almost impossible to come up with a constructive proof.
