Given an irrational number \(\xi\), consider the infimum \(\nu(\xi)\ge 1\) of the real numbers \(u\) such that \(\vert\vert q\xi\vert\vert>q^{-u}\) for all sufficiently large integer \(q>0\), where \(\vert\vert x\vert\vert\) denotes the distance of a real number \(x\) to the nearest integer. The number \(\nu(\xi)+1\) is called the irrationality exponent of \(\xi\). By restricting the set of positive integers \(q\) in the above definition to a certain infinite subset of positive integers, other type of exponents can be defined.
A classical example is when \(q\) is of the form \(d^s\) for a fixed integer \(d\), \(\vert d\vert >2\), and \(s\in \mathbb N\). Following M. A. Bennett and Y. Bugeaud [Acta Arith. 155, No. 3, 259–269 (2012; Zbl 1306.11053)], \(\nu_d(\xi)\ge 0\) is defined as the infimum of the real numbers \(u\) such that \(\vert\vert d^s\xi\vert\vert>d^{-su}\) for all sufficiently large integer \(s>0\).
In this paper, the authors compute \(\nu_d(\xi)\) when \(\xi\) is a real number related to the values of Jacobi’s triple product \[ \Pi_q(t)=\prod_{m=1}^\infty (1-q^{2m})(1+q^{2m-1}t)(1+q^{2m-1}t^{-1}) \] and of Euler’s product \[ \pi_q(t)=\prod_{m=1}^\infty (1-q^{m}t). \] Their first result is the following. Let \(a/b\in \mathbb Q\setminus \{0\}\), with \(a,b\) coprime, \(d\in \mathbb Z\) such that \(\max(\vert a\vert,\vert b\vert)<\vert d\vert\); then for all sufficiently large integer \(s>0\) and all integer \(p\), \[ \left\vert \Pi_{1/d}(a/b)-\frac{p}{d^s} \right\vert >\frac{1}{2\vert d\vert^{s(1+\varepsilon_1(s))}}, \] where \(\varepsilon_1(s)=6/\sqrt{s}+8/s\). In particular, \(\nu_{d}(\Pi_{1/d}(a/b))=0\).
Their second result deals with \(\pi_q(t)\) but only for \(t=1\). Let \(d\in \mathbb Z\setminus\{0, \pm 1\}\), \(s\) a positive integer and \(p\) an integer; then \[ \left\vert \pi_{1/d}(1)-\frac{p}{d^s} \right\vert >\frac{1}{2\vert d\vert^{s(1+\varepsilon_2(s))}}, \] where \(\varepsilon_2(s)=(3+\sqrt{1+24s}))/(2s)\). In particular, \(\nu_{d}(\pi_{1/d}(1))=0\).
The proof of the first result is based on Jacobi’s identity \[ \Pi_{1/d}(t)=\sum_{n\in \mathbb Z} d^{-n^2}t^n, \] while the proof of the second one is based on Euler’s pentagonal formula \[ \pi_{1/d}(1)=1+\sum_{n=1}^\infty (-1)^n (d^{-n(3n-1)/2}+d^{-n(3n+1)/2}). \] These identities enable the authors to construct sequences of good rational approximations to \(\Pi_{1/q}(t)\) and \(\pi_{1/d}(1)\) respectively with denominators of a special form, from which the results are deduced by standard arguments.