Number-theoretic aspects of 1D localization: "popcorn function" with Lifshitz tails and its continuous approximation by the Dedekind η

We discuss the number-theoretic properties of distributions appearing in physical systems when an observable is a quotient of two independent exponentially weighted integers. The spectral density of ensemble of linear polymer chains distributed with the law $\sim f^L$ ($0<f<1$), where $L$ is the chain length, serves as a particular example. At $f\to 1$, the spectral density can be expressed through the discontinuous at all rational points, Thomae ("popcorn") function. We suggest a continuous approximation of the popcorn function, based on the Dedekind $\eta$-function near the real axis. Moreover, we provide simple arguments, based on the "Euclid orchard" construction, that demonstrate the presence of Lifshitz tails, typical for the 1D Anderson localization, at the spectral edges. We emphasize that the ultrametric structure of the spectral density is ultimately connected with number-theoretic relations on asymptotic modular functions. We also pay attention to connection of the Dedekind $\eta$-function near the real axis to invariant measures of some continued fractions studied by Borwein and Borwein in 1993.