混合幂次丢番图逼近问题

Recently, many interesting and important results are obtained in diophantine approximation which is an main branch of number theory. At present, it is possible for us to improve and expand these results. Many diophantine approximation problems with mixed powers and mixed variables are worthy of further studying. In this project, using Davenport-Heilbronn method flexibly we will investigate the following problems relative to the diophantine approximations. 1) In some diophantine approximation problems with prime variables, under certain conditions we will study what degree of approximation results can be obtained for diophantine approximation with mixed powers and prime variables with real coefficients, such as diophantine inequalities with prime variables and mixed powers and powers of two. 2) We will consider some special diophantine inequalities with mixed powers and integer variables or prime variables, and give the best exceptional set results when inequalities have no solution. This will broaden the application of circle method, and can be considered from another angle to study some Goldbach-Waring problems. 3) We will investigate diophantine approximation with mixed powers and mixed variables. This is a completely new research field, where many characteristics of research results can be expected to get. The problems and plans of this project are profound and meaningful. We believe that our results will help extend and enrich the current research areas of diophantine approximation.

丢番图逼近作为数论历史悠久的一个重要分支，最近有许多引人注目的研究成果。目前，这些研究成果有拓展、改进的可能。很多重要的混合变量、混合幂次的丢番图逼近问题值得我们去深入研究。本项目灵活运用Davenport-Heilbronn方法研究有关丢番图逼近的下述问题：1）在某些素数变量丢番图逼近问题中，研究混合幂次素数变量带有实系数的逼近问题，例如若干素数的混合幂次方及2的若干次幂和的丢番图不等式，在某些条件下能得到何种程度的逼近结果。2）研究混合幂次的整数变量或素数变量的丢番图逼近问题，给出某些绝对值不等式无解的最佳例外集结果。这将拓宽圆法的应用，并可视作从另一个角度研究某些Goldbach-Waring型问题。3）研究混合变量的混合幂次丢番图逼近问题。这是一个崭新的研究领域，预期可以得到许多有特色的研究结果。本项目选题深刻而有意义，结果的取得将帮助拓展、丰富当前丢番图逼近研究领域。

丢番图逼近作为数论历史悠久的一个重要分支，最近有许多引人注目的研究成果。目前，这些研究成果有拓展、改进的可能。特别是很多重要的混合变量、混合幂次的丢番图逼近问题值得我们深入研究。项目运用Davenport-Heilbronn方法、均值估计研究了有关丢番图逼近的下述问题：.（1）利用Davenport-Heilbronn方法，分别研究了整数变量和素数变量的混合幂次丢番逼近问题。在给定幂次的条件下，考虑多少整数变量或素数变量，使得相应非线性型的整数部分可表示无限多素数。我们给出了众多创新结果。.（2）在Linnik-Gallagher型方程最小素数解的上界估计方面，拓展了A. Baker关于素变数方程解的上界估计问题。例如，我们研究了一个素数与三个素数的平方和的非线性方程，给出了方程在什么条件下有素数解，以及有素数解时小素数解的上界。跟踪相关研究结果，我们一直在寻求努力得到此问题的改进结果。.（3）给出了两个素变数方程在某特殊集上解的个数渐近公式。