Besicovitch, Bisection, and the normality of $0.(1)(4)(9)(16)(25)\dots$

TL;DR

This paper revisits Besicovitch's 1935 techniques and provides a new proof demonstrating the normality of the concatenated squares constant, updating and strengthening his original methods.

Contribution

It offers a novel proof of the normality of the squares concatenation constant, enhancing Besicovitch's original combinatorial approach.

Findings

Confirmed the normality of the concatenated squares constant.

Updated Besicovitch's methods for modern combinatorial proofs.

Strengthened the theoretical understanding of normality in specific constants.

Abstract

We revisit Besicovitch's 1935 paper in which he introduced several techniques that have become essential elements of modern combinatorial methods of normality proofs. Despite his paper's influence, the results he inspired are not strong enough to reprove his original result. We provide a new proof of the normality of the constant $0.(1)(4)(9)(16)(25)\dots$ formed by concatenating the squares, updating Besicovitch's methods.