On Additive Complements And Least Common Multiples

In this thesis,we mainly research on additive complements and least common multiples.Details are as follows:1.Two infinite sets A and B of positive integers are called infinite additive complements if all sufficiently large integers can be represented as a sum of elements from A and B.Let A(x)(resp.B(x))be the counting function of set A(resp.B).In 2020,Chen and Fang proved that for additive complements A and B,if A(x)B(x)-x→+∞ does not hold,then(?).In this thesis,we establish the relationship between lim sup A(2x)B(2x)/A(x)B(x)and the conclusion A(x)B(x)-x→+∞,and prove the following result:Theorem 1 For infinite additive complements A and B,if Then A(x)B(x)-x→+∞,x→+∞.Furthermore,the constants 2 and 4 cannot be improved.2.For a strictly increasing sequence(?)of positive integers,Borwein proved that(?)for any positive integer n,where the equality holds if and only if ai=2i-1(i=1,2,…,n+1).For 3 ≤ r ≤7,Qian further proved that(?)and characterized the equality,where Ur(n)depends only on r and n.In this thesis,we determine the upper bound of(?)and also characterize the terms a1,a2,…,an+r-1 such that the best upper bound is attained.We prove the following results:Theorem 2 Let r≥2 be an integer and A={a1<a2<…<ar<…} be a strictly increasing sequence of positive integers with[a1,…,ar-1]≤ar.(ⅰ)Under the condition[a1,…,ar-1]<ar,we have SA,r(n)≤1/mr-1(1-1/2n)for any positive integer n.Furthermore,in this case,the equality SA,r(n)=1/mr-1(1-1/2n)holds if and only if,a1,a2,…,ar-1 are positive divisors of mr-1 and ai=2i-r+1mr-1 for i=r,r+1,…,n+r-1.(ⅱ)Under the condition[a1,…,ar-1]=ar,we have SA,r(n)≤2/mr(1-1/2n)for any positive integer n.Furthermore,in this case,the equality SA,r(n)=2/mr(1-1/2n)holds if and only if,a1,a2,…,ar are positive divisors of mr,and ai=2i-rmr for i=r,r+1,…,n+r-1.