On Additive Complements And Complete Sequences

In this thesis, we mainly research on additive complements and complete sequences. The main results are summarized as follows.1. We call two infinite sets A and B of non-negative integers additive complements if their sum contain all sufficiently large integers. Let A（x） and B（x） be the counting functions of A and B. In 2011, Yong-Gao Chen and Jin-Hui Fang proved that for additive complements A = ｛∑εia2i,εi=0,1,…,a-1｝,B={∑εja2j-1,εj=0,1,…,a-1},we have,and A（x）B（x）-x≡1 for x = a^2k-1.In this thesis, we extend the above result and obtain the following results（Advances in Mathematics （China） ,45（2016）, 533-536）:（i ） For the above additive complements A and B （taking a = 2）, we determine all integers x such that A（x）B（x） - x = 1.（ii ） For any integers a,b with 2≤a≤b, there exist infinite additive complements A,B such that lim sup A（x）B（x）/x=2（ab-1）/ab+a-2,and A（x）B（x）-x ＝1 for infinitely many positive integers x.2. For a sequence A of nonnegative integers, let P（A） be the set of all integers which can be represented as the sum of distinct terms of A. A sequence A of nonnegative integers is called complete if P（A） contains all sufficiently large integers.For a positive real greatest integer not greater than x and a sequence S= {s1,S2,…} of positive integers. Let Us＝｛α｜Sα, is complete} . In 1995,Hegyvari proved that, if sn+1 <γsn for all integers n≥n0,where 1<Y<2,lim（Sn+1-sn）= +∞ and Us≠（?）,then μ（Us）>0,where μ（Us） is the Lebesgue measure of Us . In 2013, Yong-Gao Chen and Jin-Hui Fang proved that, if sn+1 < γsn for all integers n≥n0,where 1 <γ≤7/4 , then μ （Us） > 0, In this thesis,we prove that （Acta Math. Hungar.,148 （2016） , 211-221） : the conclusion holds for 1<γ≤（?）=1.898….