In this article we study a numeration system previously used to prove combinatorial properties in discrete geometry: the Δ -numeration. Since this system, introduced via the fully subtractive algorithm, has been seen mainly as a tool, we propose here to study it from the point of view of numeration systems. In particular, we make the link with β -numeration and Cantor real bases. We reintroduce the rewriting system introduced to calculate in Δ -numeration. This systems is based on the properties of the fully subtractive algorithm and is normalising. Finally, we study the ultimately periodic case, a special case of alternate bases, and show that the ultimately periodic words represent exactly the elements of ℚ[β ] where β is the inverse of a Pisot number.
The Heinis spectrum is the set of all pairs (α _u,β _u) such that α _u=lim inf _n→∞p_u(n)/n and β _u=lim sup _n→∞p_u(n)/n for some infinite word u. In this paper, we demonstrate that there exists a closed connected set with non-empty interior contained in . Furthermore, every point in this set can be represented as the pair (α _u,β _u) for some recurrent word u. The construction is explicit, algorithmic in nature and is based on constructing certain “Cantor sets of integers”, whose “gaps” correspond to blocks of zeros.
In this paper we introduce the notion of generalized factorization for an arbitrary submonoid M⊆ A^* , where A^* is the free monoid generated by an alphabet A, generalizing, in this way, the notion of factorization of A^* . Then we give a characterization of the free product of two submonoids of A^* in terms of unambiguous products of monoids. To do this we make use of the notion of coding partition of a set X⊆ A^+ , where A^+ is the free semigroup generated by an alphabet A. Moreover, given a coding partition of a set X⊆ A^+ , we will show how to construct a generalized factorization of X^* .
Christol’s theorem [7] characterizes algebraic formal power series over finite fields in terms of automatic sequences, establishing a fundamental link between algebraicity and computability. A refinement of this result by Bell and the author [2] provides an algebraic characterization of formal power series whose support is a sparse automatic set, showing that sparseness can be characterized in terms of certain key transformations such as the Frobenius map, multiplicative scaling, and power transformations. In this paper, we extend these results to the multivariate setting, building on Salon’s [16] generalization of Christol’s theorem. By using a characterization of sparse regular languages, we look at algebraic multivariate formal power series whose supports are sparse subsets of ℕ^d and characterize them in terms of above-mentioned transformations, providing a structural understanding of the interplay between algebraicity and sparseness in multiple variables.
Numeration systems are maps between a set of numbers and a set of words that act as representations of these numbers. One desirable property is positionality: the ability to relate positions in the words to values of the numbers. In general, positionality is hard to decide. In this article, we obtain a criterion to decide the positionality of so-called Dumont–Thomas numeration systems, arising from substitutions. Then, we particularize this criterion to some well-behaved classes of substitutions, allowing us to link the related systems to existing literature.
