Let \(K\) denote \(\mathbb Q\) or \({\mathbb Q}(i)\) and let \(O_K\) be the ring of integers of \(K\). Let \(q\) be an element of \(O_K\) with \(|q|>1\) and let \(a\) and \(\alpha\) be non-zero elemts of \(K\) such that \(\pm\alpha, -a\alpha\neq q^j\) for any positive integer \(j\). Let \[ f(z)=\prod_{j=1}^\infty g(zq^{-j}) \] where \(g(z)=\left(1+az-z^2-az^3\right).\) If \(Q,P_1,P_2,P_3,P_4\) are in \(O_K\) and \(\neq 0\), then \[ \max\{|Qf(\alpha)-P_1|,|Qf(-\alpha)-P_2|,|Qf(i\alpha)-P_3|,|Qf(-i\alpha)-P_4|\} \geq c_1|Q|^{-24-c_2(\log|Q|)^{-1/2}}, \] with effectively computable positive constants \(c_1, c_2\).
While the form of the result is very special, this is the first such irrationality result for an infinite product of this type in which the polynomial \(g\) has degree greater than 2 and only succeeds for polynomials of degree 3 of this particular shape.
The proof proceeds via the construction of an auxiliary function using an interpolation formula along the same lines as irrationality results for theta functions. The estimation of the denominators in the construction places major restrictions on how much this approach can achieve.