On some questions of Erdős and Graham about Egyptian fractions

In this paper it is proved that, for x sufficiently large, every integer m with 1 ⩽ m ⩽ [∑ 1 ⩽ n ⩽ x1 n − 9 2( log   log   x ) 2 ( 1 + o ( 1 ) )log   x]can be written as m = Σ 1≤ n ≤ x ε n / n , where ε i , = 0 or 1.