The celebrated Roth’s 3-term progression theorem provides an answer on the cardinality of subsets \(A\) of finite fields \(\mathbb{F}_p\) guarantying a nontrivial solution to equation \(a_1-a_2=a_2-a_3\) with all variables \(a_1,a_2,a_3\in A\) and not all equal. In the first part of the paper the author gives a short commented overview of results extending this result in direction of J. Bourgain et al. [Isr. J. Math. 221, No. 2, 853–867 (2017; Zbl 1420.11024)] and M. Rudnev [Combinatorica 38, No. 1, 219–254 (2018; Zbl 1413.51001)]. He proves – among others – a variant of Rudnev’s theorem on the number of ordered solution to the equation \((a_1-a_2)(a_3-a_4)=(a_5=a_6)(a_7-a_8)\) with \(a_i\in A\), \(i=1,\dots,8\), showing that the pseudorandom term dominates when \(|A|\geq p^{3/5}\). The paper contains examples showing that the proved bound are essentially sharp. He also explains some tools allowing one to make a further progress.
For the entire collection see [Zbl 1420.00048].