ON THE SIZE OF A RESTRICTED SUMSET WITH APPLICATION TO THE BINARY EXPANSION OF $sqrt{d}$

Applicable Analysis and Discrete Mathematics

article

Arturas Dubickas¹ 

[1] Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University

关键词: Sidon sequence;    quadratic irrationality;    binary expansion;    sumset;   

DOI  :  10.2298/AADM180720014D

学科分类：社会科学、人文和艺术（综合）

来源: Univerzitet u Beogradu * Elektrotehnicki Fakultet / University of Belgrade, Faculty of Electrical Engineering

【 摘 要 】

For any A ⊆ N, let U(A, N) be the number of its elements not exceedingN. Suppose that A + A has V (A, N) elements not exceeding N, wherethe elements in the sumset A + A are counted with multiplicities. Wefirst prove a sharp inequality between the size of U(A, N) and that ofV (A, N) which, for the upper limits ω(A) = lim supN→∞ U(A, N)N−1/2andσ(A) = lim supN→∞ V (A, N)N−1, implies ω(A)2 ≥ 4σ(A)/π. Then, as anapplication, we show that, for any square-free integer d > 1 and any ε > 0,there are infinitely many positive integers N such that at least (p8/π−ε)√Ndigits among the first N digits of the binary expansion of √d are equal to 1.

【 授权许可】

Unknown