Composition inverses of the variations of the Baum-Sweet sequence (Q2317874)

Let \(\mathbf{s}=(s_{n})_{n\in\mathbb{N}}\) be a \(k\)-automatic sequence, i.e., a sequence whose \(n\)-th term is generated from the base-\(k\) expansion of \(n\) using a finite automaton. A famous Christol theorem states that if \(p\) is a prime number then the sequence \(\mathbf{s}\) with values in finite field \(\mathbb{F}_{p}\) is \(p\)-automatic if and only if the ordinary generating function of the sequence \(\mathbf{s}\), say \(F(x)\), is algebraic over \(\mathbb{F}_{p}\). Let us suppose that \(s_{0}=0\) and \(s_{1}\neq 0\). Then there is \(G\in\mathbb{F}_{p}[[x]]\) such that \(F(G(x))=G(F(x))=x\). The algebraicity of \(F\) implies the algebraicity of \(G\) and thus the sequence of coefficients of \(G\), say \(\mathbf{S}=(S_{n})_{n\in\mathbb{N}}\), is also \(p\)-automatic. The sequence \(\mathbf{S}\) is called the formal inverse of the sequence \(\mathbf{s}\). In the paper the author is interested in the arithmetic properties of the formal inverse of two relatives of Baum-Sweet sequence. The original Baum-Sweet sequence is the sequence whose \(n\)-th term is equal to 0 if the binary expansion of \(n\) contains a block of \(0\)'s of odd length and \(1\) otherwise. However, the 0-th term is defined as equal to 1, so there is no formal inverse. The author modify the initial term to be 0 and consider shifts of the Baum-Sweet sequence. There are several results proved in the paper concerning formal inverse of introduced sequences. In particular, appearance of consecutive zero's, one's, increasing sequence of indices for which the sequence takes values 1 and other properties are investigated.