Avoiding a shape, and the slice rank method for a system of equations

Abstract: Fix a vector space over a finite field and a system of linear equations. We provide estimates, in terms of the dimension of the vector space, of the maximum of the sizes of subsets of the space that do not admit solutions of the system consisting of more than one point. That from above is derived by slice rank method of Tao; to obtain one from below, we define the notion of 'dominant reductions' of the system. Furthermore, by adapting a recent argument of Sauermann, we make an estimation of the maximum of the sizes of subsets that are 'W shape'-free, that means, there exist no five distinct points forming two overlapping parallelograms.