Open AccessOn the size of a restricted sumset with application to the binary expansion of √d
Author(s) -
Artūras Dubickas
Publication year - 2019
Publication title -
applicable analysis and discrete mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.69
H-Index - 26
eISSN - 2406-100X
pISSN - 1452-8630
DOI - 10.2298/aadm180720014d
Subject(s) - mathematics , combinatorics , integer (computer science) , binary number , square (algebra) , discrete mathematics , arithmetic , geometry , computer science , programming language
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.