Hankel determinants of the Cantor sequence

TL;DR

This paper derives recurrence relations for Hankel determinants of the Cantor sequence, proves their 3-automatic nature, and uses them to establish that the Cantor number has an irrationality exponent of 2.

Contribution

It introduces recurrence equations for the Hankel determinants of the Cantor sequence and demonstrates their 3-automatic property, linking them to the irrationality measure of the Cantor number.

Findings

Hankel determinants follow specific recurrence relations.

Hankel determinants form a 3-automatic double sequence.

The irrationality exponent of the Cantor number is exactly 2.

Abstract

In the paper, we give the recurrent equations of the Hankel determinants of the Cantor sequence, and show that the Hankel determinants as a double sequence is 3-automatic. With the help of the Hankel determinants, we prove that the irrationality exponent of the Cantor number, i.e. the transcendental number with Cantor sequence as its b-ary expansion, equals 2.