On infinite arithmetic progressions in sumsets

Let k be a positive integer.Denote by D_(1/k)the least integer d such that for every set A of nonnegative integers with the lower density 1/k,the set(k+1)A contains an infinite arithmetic progression with difference at most d,where(k+1)A is the set of all sums of k+1 elements(not necessarily distinct)of A.Chen and Li(2019)conjectured that D_(1/k)=k~2+o(k~2).The purpose of this paper is to confirm the above conjecture.We also prove that D_(1/k)is a prime for all sufficiently large integers k.