Abstract
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.
Highlights
In this paper, for A = {a1 < a2 < a3 < . . . } ⊆ N, where N is the set of positive integers, we will compare the quantities
The first quantity is the number of elements of the set A ∩ [1, N ], whereas the second counts the number of elements in (A + A) ∩ [1, N ], where the elements of the sumset A + A are counted with multiplicities
In all what follows we will show that by a small perturbation of the coordinates (that is, by slightly changing zi∗ so that the new vector is in Ω and satisfies (9)) we can decrease the value of M and so get a contradiction with the minimality of max(z1, z2, . . . , z2m) in Ω
Summary
The distribu√tion of digits of a given irrational number (in its decimal or binary expansion), e.g., 2, π, e, etc., is a completely open problem One expects that those numbers are normal (see, e.g., [2], [4], [10]), but the best known results are very far from this. In the proof of the optimality of the constant in Corollary 2 we use, in addition, a result of Ruzsa [12] on Sidon sequences of polynomial type. An extension of such a result would give a wider range for α for which the constant in (3) is best possible
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