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David H. Bailey; Jonathan M. Borwein; Richard E. Crandall; Carl Pomerance
On the binary expansions of algebraic numbers
Journal de théorie des nombres de Bordeaux, 16 no. 3 (2004), p. 487-518, doi: 10.5802/jtnb.457
Article PDF | Reviews MR 2144954 | Zbl 1076.11045 | 1 citation in Cedram

Résumé - Abstract

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

$$ \#(|y|, N) > CN^{1/D}$$

for a positive number $C$ (depending on $y$) and sufficiently large $N$. This in itself establishes the transcendency of a class of reals $\sum _{n \ge 0} 1/2^{f(n)}$ where the integer-valued function $f$ grows sufficiently fast; say, faster than any fixed power of $n$. By these methods we re-establish the transcendency of the Kempner–Mahler number $\sum _{n \ge 0} 1/2^{2^n}$, yet we can also handle numbers with a substantially denser occurrence of 1’s. Though the number $z = \sum _{n \ge 0} 1/2^{n^2}$ has too high a 1’s density for application of our central result, we are able to invoke some rather intricate number-theoretical analysis and extended computations to reveal aspects of the binary structure of $z^2$.

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