Note on powers of 2 in sumsets

Let n ≥ 2 be an integer. Let A be a subset of [ 0 , n ] with 0 , n ∈ A . Assume the greatest common divisor of all elements of A is 1. Let k be an odd integer and s = k − 1 2 . Then, we prove that when 3 ≤ k ≤ 11 and | A | ≥ 7 s + 3 ( s + 1 ) ( 7 s + 4 ) ( n − 2 ) + 2 , there exists a power of 2 which can be represented as a sum of k elements (not necessarily distinct) of A . But when k ≥ 13 , the above constraint should be changed to | A | ≥ s + 1 s 2 + 2 s + 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.