A note on exponential sums over subgroups of Zp2∗ and their applications

Let G be a subgroup of Zq∗, and #G=t, set S(G)=maxa∈Zq∗|∑x∈Geq(ax)|, and Tk(G):=#{(x1,x2,…,x2k):x1+⋯+xk=xk+1+⋯+x2k(modq)xi∈G}. As q=p2, we obtain the general cases of Tk(G), then one can easily obtain the nontrivial bound of S(G) as p2/3+ϵp7/10 from Malykhin (2005) [12]. On the other hand, it is known J. Bourgain obtain the nontrivial bound for t>pϵ with arbitrary ϵ>0 by the sum-product method, however his bounds not be explicit. We also give some connections between Tk(G) and Erdös-Szemerédi Conjecture.