Squarefree numbers in sumsets

We solve a problem of Filaseta by proving, that if N is sufficiently large, A ⊆ [N], |A| > N/9 and A + Adoes not contain any squarefree integer then all elements of A are congruent to 0 (mod 4) or 2 (mod 4). In order to show the main result we characterize the structure of all dense sets, whose elements sum to no squarefree number.