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On the binary expansions of algebraic numbers

David H. BaileyJonathan M. BorweinRichard E. CrandallCarl Pomerance — 2004

Journal de Théorie des Nombres de Bordeaux

Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1’s in the binary expansions of real algebraic numbers. A central result is that if a real y has algebraic degree D > 1 , then the number # ( | y | , N ) of 1-bits in the expansion of | y | through bit position N satisfies # ( | y | , N ) > C N 1 / D for a positive number C (depending on y ) and sufficiently large N . This in itself establishes the transcendency of a class of...

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