加法数论中的例外集问题及Linnik问题

加法数论问题是数论中基础且重要的研究课题，受到了国内外众多学者的重视，对数论的发展发挥了重要的作用。本项目拟研究加法数论中的例外集问题和Linnik问题。现有技术下，在许多加法问题上很难取得重要进展，研究这些问题的例外集问题及Linnik问题就成为了逼近这些问题的重要研究途径。具体来说，将应用圆法、筛法、三角和等工具，研究一类Waring问题或Waring-Goldbach问题的例外集问题和Linnik问题；研究推广型的Linnik问题；探索运用例外集问题的研究工具考察Linnik问题，建立两者之间的联系。本项目的预期成果将为一些加法问题提供新的研究思路。

The study of additive number theory is a fundamental and significant research subject and plays an extremely important role in the development of number theory. Since it is difficult to get good results by now in the research of additive number theory problems, the study of exceptional set and Linnik problem plays important roles in those problems. In this project, we intent to study the exceptional set and the Linnik problem of additive number theory. For detail, we try to apply circle method, sieve method, trigonometric sum, etc. to discover the exceptional set and Linnik problem of a class of Waring problem or Waring-Goldbach problem; study the generalized Linnik problem; study Linnik problem by the tool of exceptional set problem, discover the relation between those two problems. The exceptional results of this project will provide some important mathematical methods for the study of additive number theory problem.