On the binary expansions of algebraic numbers

Bailey, David H.; Borwein, Jonathan M.; Crandall, Richard E.; Pomerance, Carl

Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 3, pp. 487-518.

En combinant des concepts de théorie additive des nombres avec des résultats sur les développements binaires et les séries partielles, nous établissons de nouvelles bornes pour la densité de $1$ dans les développements binaires de nombres algébriques réels. Un résultat clef est que si un nombre réel $y$ est algébrique de degré $D>1$, alors le nombre $\#\left(\right|y|,N)$ de $1$ dans le développement de $\left|y\right|$ parmi les $N$ premiers chiffres satisfait

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

avec un nombre positif $C$ (qui dépend de $y$), la minoration étant vraie pour tout $N$ suffisamment grand. On en déduit la transcendance d’une classe de nombres réels ${\sum }_{n\ge 0}1/2^{f\left(n\right)}$ quand la fonction $f$, à valeurs entières, croît suffisamment vite, disons plus vite que toute puissance de $n$. Grâce à ces méthodes on redémontre la transcendance du nombre de Kempner–Mahler ${\sum }_{n\ge 0}1/2^{2^n}$ ; nous considérons également des nombres ayant une densité sensiblement plus grande de $1$. Bien que le nombre $z={\sum }_{n\ge 0}1/2^{n^2}$ ait une densité de $1$ trop grande pour que nous puissions lui appliquer notre résultat central, nous parvenons à développer une analyse fine de théorie des nombres avec des calculs étendus pour révéler des propriétés de la structure binaire du nombre $z^2$.

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 $\#\left(\right|y|,N)$ of 1-bits in the expansion of $\left|y\right|$ through bit position $N$ satisfies

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

for a positive number $C$ (depending on $y$) and sufficiently large $N$. This in itself establishes the transcendency of a class of reals ${\sum }_{n\ge 0}1/2^{f\left(n\right)}$ 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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