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

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 ω(A) = lim sup N →∞ U (A, N )N −1/2 and σ(A) = lim sup N →∞ V (A, N )N −1 , implies ω(A) 2 ≥ 4σ(A)/π. Then, as an application, 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 ( 8/π−ε) √ N digits among the first N digits of the binary expansion of √ d are equal to 1.

Gaps in a Sumset of a Polynomial With Itself

“…In principle, the next result can be verified by a direct calculation. However, we will give a much shorter proof applying a recent result on the cardinality of sumsets from [7]. (Throughout, Γ(x) = ∞ 0 t x−1 e −t dt is the gamma function.)…”

“…we first take a real number t satisfying (7) {t} = {(X/a) (Here and in (9), {y} stands for the fractional part of y ∈ R.) With this choice, as X → ∞, the number u = (X/a) 1/d − t is a positive integer and ( 8)…”

Lacunary sequences in analysis, probability and number theory

“…Many such open problems "for the next millennium" are contained in Harman's survey article [137]; see also [33]. However, it is quite clear that the mathematical machinery which would be necessary to prove the normality of √ 2 or other such constants is completely lacking; compare the rather deplorable current state of knowledge on the binary digits of √ 2 as given in [34,98,217]. A small spark of hope is provided by the very remarkable formulas of Bailey, Borwein and Plouffe (now widely known as BBP formulas), which allow to calculate deep digits of π (and other constants) without the need of computing all previous digits.…”