Character sums over shifted powers

We study character sums over shifted powers modulo a prime $p$. Such sums can be viewed as generalizations of character sums over shifted multiplicative subgroups. We obtain some new results on upper estimates for absolute value of these sums. The case when the cardinality of subgroup is less than $\sqrt{p}$, it is a question of non-trivial upper bounds for such sums that remains open and is unsolved today. It was proposed by J. Burgain and M. Ch. Chang in the review of 2010. Nevertheless, some intermediate results were achieved by Professor K. Gong, who established non-trivial estimates of such sums in the case when the subgroup is much larger than $\sqrt{p}$. In this paper, we obtain some new results on the upper bound for the absolute value of the generalization of such sums, which are incomplete sums of character sums over shifted subgroups. Two proofs of the main result are given. The first one is based on reduction of this sum to the well-known estimate of A. Weil and the method of smoothing such sums. The method of estimating the incomplete sum through the full one is also applied. One result of M. Z. Garaev is also used. The second proof is based on the original idea of I. M. Vinogradov. This approach was proposed to refine the known inequality of Poya–Vinogradov and uses in its essence some geometric and combinatorial ideas. The second proof is not fully presented. We only prove a key statement, and for the rest of the calculations we refer the reader to the initial work of I. M. Vinogradov.
Bibliography: 15 titles.