Roth's theorem on arithmetic progressions

Roth's theorem, established by mathematician Klaus Roth in 1951, is a significant result in the field of number theory, particularly in the study of arithmetic progressions and additive combinatorics. The theorem specifically deals with the distribution of rational approximations to irrational numbers. In its classical form, Roth's theorem states that if \(\alpha\) is an irrational number, then it cannot be well-approximated by rational numbers in a very precise way.