On the size of a restricted sumset with application to the binary expansion of √d

For any A ? N, let U(A,N) be the number of its elements not exceeding N. Suppose that A + A has V (A,N) elements not exceeding N, where the elements in the sumset A + A are counted with multiplicities. We first prove a sharp inequality between the size of U(A,N) and that of V (A,N) which, for the upper limits ?(A) = lim supN?? U(A,N)N-1/2 and ? (A) = lim sup N?? V (A,N)N-1, implies ?(A)2 ? 4 ? (A)/?. Then, as an application, we show that, for any square-free integer d > 1 and any ? > 0, there are infinitely many positive integers N such that at least (?8/ ?- ?) ?N digits among the first N digits of the binary expansion of ?d are equal to 1.