Exponential sums and the abelian group problem

We give new estimates for multiple exponential sums, which infers $$A(x)=C_1x+C_2x^{1/2}+C_3x^{1/3}+O(x^{1/4}e^{V(x)}), V(x)=\frac{1}{\sqrt{3}}(L\log{L})^{1/...