Note on integer powers in sumsets

Let k,m,n⩾2k,m,n⩾2 be integers. Let A   be a subset of {0,1,…,n}{0,1,…,n} with 0∈A0∈A and the greatest common divisor of all elements of A is 1. Suppose that|A|>1l+12-klmn+2l,where l=⌈k/m⌉l=⌈k/m⌉. We prove that if m⩾3m⩾3, or m=2m=2 and k even, then there exists a power of m which can be represented as a sum of k elements (not necessarily distinct) of A.