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