Banach's Indicatrix Reloaded

TL;DR

This paper refines Banach's classical theorem by linking a function's smoothness, measured via integral modulus of continuity, to the structure of its superlevel sets, using a continuum incidence approach.

Contribution

It provides a refined version of Banach's theorem and addresses a question posed by Garsia--Sawyer through a novel incidence-based method.

Findings

Refined relation between function smoothness and superlevel set structure.

Established a continuum incidence framework for analyzing open set regularity.

Answered an open question of Garsia--Sawyer.

Abstract

Banach famously related the smoothness of a function to the size of its level sets. More precisely, he showed that a continuous function is of bounded variation exactly when its "indicatrix" is integrable. In a similar vein, we connect the smoothness of the function -- measured now by its integral modulus of continuity -- to the structure of its superlevel sets. Our approach ultimately reduces to a continuum incidence problem for quantifying the regularity of open sets. The pay off is a refinement of Banach's original theorem and an answer to a question of Garsia--Sawyer.