Random Difference Sets In Z_N~d

The research of arithmatic structures in random sets is important issue.In this thesis, we will investigate the structure of dense random sets,and prove that difference sets con-tain a power with high probability.The proofs rely on a restriction-type Fourier analytic arguments due to Green and Tao.There are mainly two results in the thesis:Part1:Beingness of k powerLet0<δ≤1,and N≥5be a prime integer.Letf:ZdNâ†'[0,1] be a bounded function such that Given k=(k1,k2,…,kd),rk=(r1k1,r2k2,…,rdkd),Then we have where c(δ) depends on δ.Part2:k-th power in random difference setsLet k≥3is a integer,suppose that W(?)ZNd such that the events x∈ZNd,where x ranges over W are independent and have probability p=p(Nd)∈(cN-θd,1]with0<θ<θk,θk=1/10·(6k-1),θk depends on k.Let α>0. Then the statementfor every set A (?) W with|A|≥α|W|,there are x,y∈A,n∈Nd for some n∈Nd such that x-y=nk.