Some results about entropy and divergence in number theory

TL;DR

This paper explores inequalities involving entropy and divergence in number theory, revealing extremal cases and applying these concepts to algebraic integers, ideals, and probability distributions on infinite trees.

Contribution

It introduces new inequalities relating entropy and divergence for numbers and ideals, and applies these to Schur { ext{-}}groups and class field towers.

Findings

Minimal entropy occurs at sharp localization

Maximal entropy occurs at equidistribution

Entropies of distributions on infinite trees are determined

Abstract

We obtain inequalities involving the entropy of a positive integer and the divergence of two positive integers, respectively the entropy of an ideal and the divergence of two ideals in a ring of algebraic integers. Among the important results, we show that the minimal entropy arises for sharp localization, and the maximal entropy occurs for equidistribution. We also study other interesting estimates of entropy and divergence for numbers and for ideals. Finally, we determine the entropies of probability distributions on infinite trees of Schur {\sigma}-groups, which are realized by 3-class field tower groups of imaginary quadratic number fields.