A note on exponential sums over subgroups of Z(p2)* and their applications

作者：Zhou, Binbin^*

来源：Journal of Number Theory, 2010, 130(11): 2467-2479.

DOI：10.1016/j.jnt.2010.05.004

• Exponential sums
• Erdos-Szemeredi Conjecture
• Subgroup

Let G be a subgroup of Z(q)(*). and #G = t, set S(G) = max(a is an element of Zq*)(vertical bar)Sigma(x is an element of G) e(q)(ax)vertical bar. and T-k(G):= #1(x(1), x(2), ..., x(2k)): x(1) + ... + x(k) = x(k+1) + ... + x(2k) (mod q) x(1) is an element of G). As q = p(2), we obtain the general cases of T-k(G), then one can easily obtain the nontrivial bound of S(G) as p(2/3+epsilon) <t <= p, which improves t > p(7/10) from Malykhin (2005) [12]. On the other hand, it is known J. Bourgain obtain the nontrivial bound for t > p(epsilon) with arbitrary epsilon > 0 by the sum-product method, however his bounds not be explicit. We also give some connections between T-k(G) and Erdos-Szemeredi Conjecture.

• 出版日期2010-11
• 单位湖南师范大学