A topology for Engel expansions: evaluation and digit coding maps
Min Woong Ahn
TL;DR
This work develops a canonical topological framework for Engel expansions by endowing the Engel digit space $\Sigma_E$ with the product topology and analyzing the evaluation map $\varphi$ and the digit coding map $f$ under a nonterminating convention. It proves that $\varphi$ is globally Lipschitz and that $f$ is continuous precisely at irrationals (and one-sided at rationals), establishing a homeomorphism between $\Sigma_E^{\mathrm{irr}}$ and $(0,1]\setminus\mathbb{Q}$. These continuity properties streamline Baire category arguments for digit-based functions, showing that certain divergence sets, including the set where the Engel convergence exponent $\lambda(x)=\infty$, are comeager in $(0,1]$. The framework clarifies the correspondence between cylinders and fundamental intervals and highlights that the Engel topology renders continuity and Baire-category statements largely tautological, with potential applications to related expansions such as Pierce.
Abstract
We develop a topological framework for Engel expansions that treats both directions of the correspondence between points of $(0,1]$ and nondecreasing digit sequences. We endow the sequence space with the product topology to study the evaluation map, and we fix a nonterminating digit algorithm to study the digit coding map. We also record the correspondence between cylinder sets and fundamental intervals, and give an application to Baire category results for functions of the digits.