Note on powers of 2 in sumsets

Let n≥2n≥2 be an integer. Let AA be a subset of [0,n][0,n] with 0,n∈A. Assume the greatest common divisor of all elements of AA is 1. Let kk be an odd integer and s=k−12. Then, we prove that when 3≤k≤113≤k≤11 and |A|≥7s+3(s+1)(7s+4)(n−2)+2, there exists a power of 2 which can be represented as a sum of kk elements (not necessarily distinct) of AA. But when k≥13k≥13, the above constraint should be changed to |A|≥s+1s2+2s+2(n−2)+2. In the present paper, we generalize the results of Pan and Lev, and obtain a non-trivial progress towards a conjecture of Pan.