Undecidable arithmetic properties of solutions of Fredholm integral equations (Q2116752)

The author considers the problem of determining whether a value of a transcendental function \(f(z)\) is rational or algebraic for a fixed value of \(z\). Such an algorithm exists for a class of functions called \(E\)-functions due to results of \textit{B. Adamczewski} and \textit{T. Rivoal} [Bull. Lond. Math. Soc. 50, No. 4, 697--708 (2018; Zbl 1450.11076)]. An \(E\)-function function \(f(z)\) satisfies the formula \(f(z)=\sum_{k=0}^{\infty}a_kz^k\), \(a_k \in \overline{\mathbb Q}\), where \(\overline{\mathbb Q}\) is an algebraic closure of \(\mathbb Q\), and is an entire function such that the coefficients \(a_k\) of its series satisfy certain arithmetic conditions while \(f(z)\) satisfies a certain linear differential equation. The author shows that for a class of functions satisfying a Fredholm integral equation (as opposed to a differential equation as was the case above), there is no algorithm to determine whether the value of the function at 0 is rational. This conclusion is obtained by simulating iterations of a generalized Collatz function and using a result of [\textit{J. H. Conway}, in: Proc. Number Theory Conf., Univ. Colorado, Boulder, 49--52 (1972; Zbl 0337.10041)] that the Generalized Collatz Problem is undecidable.