On a generalization of a theorem of Erdős and Fuchs

Let A = { a 1 , a 2 , … } ( a 1 < a 2 < ⋯ ) be an infinite sequence of nonnegative integers, let k ≥ 2 be a fixed integer and denote by r k ( A , n ) the number of solutions of a i 1 + a i 2 + ⋯ + a i k ≤ n . Montgomery and Vaughan proved that r 2 ( A , n ) = c n + o ( n 1 / 4 ) cannot hold for any constant c > 0 . In this paper, we extend this result to k > 2 .