An improvement of a theorem of Erdős and Sárközy

Let a 1 < a 2 < … be an infinite sequence of positive integers and denote by R 2 ( n ) the number of solutions of n = a i + a j . P. Erdős and A. Sárközy proved that if g ( n ) is a monotonically increasing arithmetic function with g ( n ) → +∞ and g ( n ) = o ( n (log n ) −2 ) then | R 2 ( n ) − g ( n )| = o (√ g ( n )) cannot hold. We will show that for any ɛ > 0, the inequality | R 2 ( n ) − g ( n )| ≤ (1 − ɛ )√ g ( n ) cannot hold from a certain point on