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We discuss the number-theoretic properties of distributions appearing inphysical systems when an observable is a quotient of two independentexponentially weighted integers. The spectral density of ensemble of linearpolymer chains distributed with the law $\sim f^L$ ($0<f<1$), where $L$ is thechain length, serves as a particular example. At $f\to 1$, the spectral densitycan be expressed through the discontinuous at all rational points, Thomae("popcorn") function. We suggest a continuous approximation of the popcornfunction, based on the Dedekind $\eta$-function near the real axis. Moreover,we provide simple arguments, based on the "Euclid orchard" construction, thatdemonstrate the presence of Lifshitz tails, typical for the 1D Andersonlocalization, at the spectral edges. We emphasize that the ultrametricstructure of the spectral density is ultimately connected with number-theoreticrelations on asymptotic modular functions. We also pay attention to connectionof the Dedekind $\eta$-function near the real axis to invariant measures ofsome continued fractions studied by Borwein and Borwein in 1993.

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