Arithmetic progressions in multiplicative groups of finite fields (Q1686398)

Let \(r\) be a positive integer, \(p\) be a sufficiently large prime and \(\delta>0\). Let \(G\) be a subgroup of the multiplicative group of a finite field with \(p\) elements. Let the size of \(G\) is larger than \(p^{1-\kappa}\) where \(\kappa=1/(r2^{r+1})\). Then the author proves that any subset \(A\) of \(G\) such that \(|A|>\delta|G|\) contains a non-trivial arithmetic progressions of length \(r\). The key ingredient of the proof is the \textit{B. Green} and \textit{T. Tao} [Ann. Math. (2) 167, No. 2, 481--547 (2008; Zbl 1191.11025)] extension of the theorem of Szemerédi for pseudorandom weights. The author also gives a construction of large multiplicative groups which not containing non-trivial arithmetic progressions of length \(r\).