A note on exponential sums over subgroups of and their applications

Abstract: Let G be a subgroup of , and , set , and . As , we obtain the general cases of , then one can easily obtain the nontrivial bound of as , which improves from Malykhin (2005) . On the other hand, it is known J. Bourgain obtain the nontrivial bound for with arbitrary by the sum-product method, however his bounds not be explicit. We also give some connections between and Erdös–Szemerédi Conjecture. [Copyright &y& Elsevier]