Hankel determinants of the Cantor sequence

Abstract

Let τ be the substitution 1 → 101 and 0 → 1 on the alphabet {0, 1}. The fixed point of τ leading by 1, denoted by s, is a Sturmian sequence. We first give a characterization of s using f-representation. Then we show that the distribution of zeros in the determinants induces a partition of integer lattices in the first quadrant. Combining those properties, we give the explicit values of the Hankel determinants Hm,n of s for all m ≥ 0 and n ≥ 1. Contents 1. Introduction 1 2. Some properties of the sequence s 3 3. Partition of the lattice 11 4. Relations of Hankel determinants 12 5. Evaluating the Hankel determinants 20 6. Proof of Theorem 1.1, 1.2 and 1.3 24 Acknowledgement 25 References 25

On the irrationality exponent of the regular paperfolding numbers

“…Using Bugeaud's method, in 2012, Coons [12] proved that the irrationality exponent of the sum of the reciprocals of the Fermat numbers is 2. Recently, Wen and Wu [23] showed that the irrationality exponents of the Cantor real numbers are exactly 2 in the same way.…”

“…In 2012, with the help of C++ program, Coons and Vrbik [11] showed that the Hankel determinant H n (F ) = 0 of the regular paperfolding sequence for n ≤ 2 13 + 3. In 2013, Wen and Wu [23] investigated the Hankel determinants of the Cantor sequence which is the fixed point of the endomorphism θ : 1 → 101, 0 → 000. They also proved that the Hankel determinants of the Thue-Morse sequence are all nonzero.…”