代数整数环上表示函数与和集问题的研究

本项目主要致力于研究代数整数环上的表示函数与和集问题，分为以下三个方面：(1)考虑代数整数环中的例外单位，研究代数整数环的和集性质、加法结构等，同时在此基础上构造一类代数整数环上的加法基；(2)结合正则卷积的研究结果，定义正则例外单位，研究正则例外单位在不具有群结构的集合中的计数问题与加法结构问题，进而研究正则条件下的一系列加法数论问题；(3)利用代数整数环上Ramanujan和以及同余方程的研究成果，探讨代数整数环的理想的和集性质以及相关表示函数问题。申请人已积累与本项目相关的研究成果，前期工作发表在Acta Arith.,Ramanujan J.,Int. J. Number Theory,Acta Math.Hungar.等期刊上，预期结果将丰富代数整数环中的和集及表示函数等研究领域的内容。

This project is mainly devoted to the sumsets and related representation functions in the rings of algebraic integers, which are divided into the following three aspects. Firstly, we define a kind of generalized exceptional units in the rings of algebraic integers, study its properties of sumsets and additive structure etc., and construct a kind of additive bases of the rings of algebraic integers. Secondly, based on the results of regular convolutions, we define the regular exceptional units, and study the counting problem and additive structure problem of regular exceptional units in sets without group structure,and then research a series of problems of additive number theory about the regular conditions. Finally,by the Ramanujan sum and congruence equation in the rings of algebraic integers,we discuss the sumsets of ideals of the rings of algebraic integers and related problems. The applicant has accumulated research results related to this project, previous studies have been published in Acta Arith.,Int. J. Number Theory, Acta Math. Hungar and so on. It is expected that the results will enrich the research fields of sumsets and representation functions.