The Study Of Some Problems Related To Sumsets

Let N be set of all nonnegative integers. For positive integer h > 2 and finite integer set A = {a0, a1,..., ak-1}, let l(A) denote the difference of the largest and the smallest elements of A. Define hA as the set of all sums of h not necessarily distinct elements of A, h^A as the set of all sums of h distinct elements of A. For positive integer r, define For finite integer set A, the cardinality problem of hA and h^A is an important sub-ject of sumsets. For infinite set A = {a1, a2,...}(?)N, let D(A) = gcd{ak+1-ak k = 1,2,...}. We say that A is an asymptotic basis of order h, if every sufficiently large integer can be expressed as a sum of at most h integers in A.The main work of this thesis is divided into two parts. In the first part, for finite integer set A, we give the best lower bound for the cardinality of 3^A, and we discuss the cardinality of h^A and h(r)A under certain conditions, we obtain the following results:(a) If l(A) ? 2|A|-5, then |3^A|? 2|A| + l(A) - 7.(b) For integer h?3, if l(A) ? 2|A|-2h+1, then |h^A|?(h-1)|A|+l(A)-h2+2.(c) Let t?1 and h?t + 1. For k? h + t +1, let where 0 ?ri?k - t - 2,i=1,…,t, then(d) Leth, r, m, t be positive integers with r?2 and h = mr +t, 0?t?r-1. If l(A)?2|A|-2(m +t)-1, then(?) |h(r)A|?(h-m-r+2)|A|+(m+r-2)l(A)-3(h-r)+1 when r < h?2r.(?) |h(r)A|?(h-1)|A|+rl(A)-(m2-1)r-(2m + 1)t+2 when 2r < h ?r|A|.In the second part, we give a necessary and sufficient condition of an infinite integer set to be an asymptotic basis.Let A = {a1 < a2 < ...} be a set of infinite nonnegative integers and the difference of consecutive integers of A is bounded. Then A is an asymptotic basis if and only if there exists an integer a ? A such that gcd(a, D(A)) = 1.