Countable $\mathcal {L}$-structures $\mathcal {N}$ whose isomorphism class supports a permutation invariant probability measure in the logic action have been characterized by Ackerman–Freer–Patel to be precisely those $\mathcal {N}$ which have no algebraicity. Here we characterize those countable $\mathcal {L}$-structures $\mathcal {N}$ whose isomorphism class supports a quasi-invariant probability measure. These turn out to be precisely those $\mathcal {N}$ which are not “highly algebraic”—we say that $\mathcal {N}$ is highly algebraic if outside of every finite F there is some b and a tuple $\bar {a}$ disjoint from b so that b has a finite orbit under the pointwise stabilizer of $\bar {a}$ in $\mathrm {Aut}(\mathcal {N})$. As a byproduct of our proof we show that whenever the isomorphism class of $\mathcal {N}$ admits a quasi-invariant measure, then it admits one with continuous Radon–Nikodym cocycles.

Hilbert schemes are an object arising from geometry and are closely related to physics and modular forms. Recently, there have been investigations from number theorists about the Betti numbers and Hodge numbers of the Hilbert schemes of points of an algebraic surface. In this paper, we prove that Göttsche's generating function of the Hodge numbers of Hilbert schemes of $n$ points of an algebraic surface is algebraic at a CM point $\tau$ and rational numbers $z_1$ and $z_2$. Our result gives a refinement of the algebraicity on Betti numbers.

We study algebraicity and transcendency of certain basic special values of the double sine functions due to Hölder and Shintani by employing the zeta regularized product expressions.