On the rational approximation of the sum of the reciprocals of the Fermat numbers

Abstract

Let $\C{G}(z):=\sum_{n=0}^\infty z^{2^n}(1-z^{2^n})^{-1}$ denote the generating function of the ruler function, and $\C{F}(z):=\sum_{n=0}^\infty z^{2^n}(1+z^{2^n})^{-1}$; note that the special value $\C{F}(1/2)$ is the sum of the reciprocals of the Fermat numbers $F_n:=2^{2^n}+1$. The functions $\C{F}(z)$ and $\C{G}(z)$ as well as their special values have been studied by Mahler, Golomb, Schwarz, and Duverney; it is known that the numbers $\C{F}(\ga)$ and $\C{G}(\ga)$ are transcendental for all algebraic numbers $\ga$ which satisfy $0<\ga<1$. For a sequence $\mathbf{u}$, denote the Hankel matrix $H_n^p(\mathbf{u}):=(u({p+i+j-2}))_{1\leqslant i,j\leqslant n}$. Let $\ga$ be a real number. The {\em irrationality exponent} $\mu(\ga)$ is defined as the supremum of the set of real numbers $\mu$ such that the inequality $|\ga-p/q|<q^{-\mu}$ has infinitely many solutions $(p,q)\in\B{Z}\times\B{N}.$ In this paper, we first prove that the determinants of $H_n^1(\mathbf{g})$ and $H_n^1(\mathbf{f})$ are nonzero for every $n\geqslant 1$. We then use this result to prove that for $b\geqslant 2$ the irrationality exponents $\mu(\C{F}(1/b))$ and $\mu(\C{G}(1/b))$ are equal to 2; in particular, the irrationality exponent of the sum of the reciprocals of the Fermat numbers is 2.