Automatic sets of rational numbers

Let k be an integer ≥ 2, and let N = {0, 1, 2, . . .} denote the set of non-negative integers. Let Σ k = {0, 1, . . . , k -1} be an alphabet of k letters. Given a word w = a 1 a 2 • • • a t ∈ Σ * k , we let [w] k denote the integer that it represents in base k; namely,

where (as usual) an empty sum is equal to 0. For example, [0101011] 2 = 43. Note that in this framework, every element of N has infinitely many distinct representations as words, each having a certain number of leading zeroes. Among all such representations, the one with no leading zeroes is called the canonical representation; it is an element of C k := {ǫ} ∪ (Σ k -{0})Σ * k . For an integer n ≥ 0, we let (n) k denote its canonical representation. Thus, for example, (43) 2 = 101011. Note that ǫ is the canonical representation of 0.

Given a language L ⊆ Σ * k , we can define the set of integers it represents, as follows:

We now recall a well-studied concept, that of k-automatic set (see, e.g., [8,9,2]):

Definition 1. We say that a set S ⊆ N is k-automatic if there exists a regular language

Many properties of these sets are known. For example, it is possible to state an equivalent definition involving only canonical representations:

To see the equivalence of Definitions 1 and 2, note that if L is a regular language, then so is the language L ′ obtained by removing all leading zeroes from each word in L.

A slightly more general concept is that of k-automatic sequence. Let ∆ be a finite alphabet. Then a sequence (or infinite word) (a n ) n≥0 over ∆ is said to be k-automatic if, for every c ∈ ∆, the set of fibers F c = {n ∈ N : a n = c} is a k-automatic set of natural numbers. Again, this class of sequences has been widely studied [8,9,2]. The following result is well-known [10]: Theorem 3. The sequence (a n ) n≥0 is k-automatic if and only if its k-kernel, the set of its subsequences K = {(a k e n+f ) n≥0 : e ≥ 0, 0 ≤ f < k e }, is finite.

In previous papers [23,22], the second author extended the notion of k-automatic sets over N to subsets of Q ≥0 , the non-negative rational numbers. The motivation was to study the “critical exponent” of automatic sequences. In this paper, we will obtain some basic results about this new class. Our principal results are Theorem 17 (characterizing those k-automatic sets of rationals consisting entirely of integers), Theorem 22 and Corollary 23 (showing that the class of k-automatic sets of rationals is not closed under intersection or complement), Theorem 29 (showing that it is decidable if a k-automatic set of rationals is infinite), and Theorem 33 (showing that it is decidable if a k-automatic set of rationals equals N).

The class of sets we study has some similarity to another class studied by Even [11] and Hartmanis and Stearns [13]; their class corresponds to the topological closure of a small subclass of our k-automatic sets, in which the possible denominators are restricted to powers of k.

Yet another model of automata accepting real numbers was studied in [1,4,5,6]. In this model real numbers are represented by their (possibly infinite) base-k expansions, and the model of automaton used is a nondeterministic Büchi automaton. However, even when restricted to rational numbers, this model does not define the same class of sets, as we will show below in Corollary 25. A preliminary version of this paper appeared in [17].

A natural representation for the non-negative rational number p/q is the pair (p, q) with q = 0. Of course, this representation has the drawback that every element of Q ≥0 has infinitely many representations, each of the form (jp/d, jq/d) for some j ≥ 1, where d = gcd(p, q). We might try to ensure uniqueness of representations by considering only “reduced” representations (those in “lowest terms”), which amounts to representing p/q by the pair (p/d, q/d) where d = gcd(p, q). In other words, the only valid pairs are (p, q) with gcd(p, q) = 1. However, the condition gcd(p, q) = 1 cannot be checked, in general, by finite or even pushdown automata -see Remark 26 below -and it is not currently known if it is decidable whether a given regular language consists entirely of reduced representations (see Section 7). Furthermore, insisting on only reduced representations means that some “reasonable” sets of rationals, such as {(k m -1)/(k n -1) : m, n ≥ 1} (see Corollary 24), have no representation as a regular language. For these reasons, we allow the rational number p/q to be represented by any pair of non-negative integers (p ′ , q ′ ) with p/q = p ′ /q ′ . Next, we need to see how to represent a pair of integers as a word over a finite alphabet. Here, we follow the ideas of Salon [18,19,20]. Consider the alphabet Σ 2 k . A finite word w over Σ 2 k can be considered as a sequence of pairs w 18). We also define ×, which allows us to join two words w, x ∈ Σ * k of the same length to create a single word w × x ∈ (Σ 2 k ) * whose π 1 projection is w and π 2 projection is x.

In this framework

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