An Extremal Problem On Anisotropic Corners In Finite Field

Firstly, the concept of corner is general extended. For x.y.d1.d2∈IF(?). the corner is redefined as a triple in the form ((x.y)(x+d1.y)(x.y+d2)).and it is anisotropic when d1≠0.d2≠0, and d1≠d2.After that. the proposition 2.1 generalised von Neumann, the proposition 2.2 density increment on a product set. and the proposition 2.3 uniformising a product set arc given. Thus we can get the dichotomy between structure and randomness theorem. Then we empoly it working out the theorem 2: (?)ε>0.(?)C0.ε> 0.s.t.(?)n∈N+, where r∠(IFnp)(?)max{|A|:A(?)IFnp×IFnp.A dose not contain anisotropic corners}.This theorem is a generalization of Roth theorem on two dimension space.