A Diophantine upper bound on the dimensions of certain spaces of holonomic functions was the main ingredient in our proof with Calegari and Tang of the 'unbounded denominators conjecture' (presented by Tang in last year's number theory seminar) from the theory of non-congruence and vector-valued modular forms. In this talk, I will report on our sequel joint work-in-progress where we extend the scope of these arithmetic holonomy bounds to beneath the framework of finite index subgroups of SL_2(Z) and onto the arithmetic theory of certain periods appearing as Apery limits for local systems on the triply punctured projective line. Applications include irrationality proofs, with quantitative bad approximability measures, of the 2-adic realization of ζ(5), the archimedean period L(2,χ−3)−π(log3)/(33‾√), and the products of two logarithms log(1−1/m)log(1−1/n) for arbitrary integer pairs n,m with 0<|1−m/n|<ϵ0, where ϵ0 is some positive absolute constant. As a byproduct, we find an arithmetic characterization of the logarithm function