Complexity of Hartman sequences par

Let T : x 7→ x + g be an ergodic translation on the compact group C and M ⊆ C a continuity set, i.e. a subset with topological boundary of Haar measure 0. An infinite binary sequence a : Z 7→ {0, 1} defined by a(k) = 1 if T (0C) ∈ M and a(k) = 0 otherwise, is called a Hartman sequence. This paper studies the growth rate of Pa(n), where Pa(n) denotes the number of binary words of length n ∈ N occurring in a. The growth rate is always subexponential and this result is optimal. If T is an ergodic translation x 7→ x+ α (α = (α1, . . . , αs)) on T and M is a box with side lengths ρj not equal αjZ + Z for all j = 1, . . . , s, we show that limn Pa(n)/n = 2 ∏s j=1 ρ s−1 j . Acknowledgement. The authors thank the Austrian Science Foundation FWF for supporting the research for this paper through projects no. S8312 and no. S8302. 1. Motivation and Notation The notion of a Hartman sequence has recently been introduced and studied (cf. [5], [10], [12]). It can be seen as a generalisation of the notion of a Sturmian sequence. Sturmian sequences (and their close relatives, the Beatty sequences) are very interesting objects, as well from the combinatorial, the number theoretic and the dynamical point of view. Let us sketch two approaches. 348 Christian Steineder, Reinhard Winkler Consider sequences a = (ak) of two symbols, say 0 and 1, where k runs through the set Z of all integers or N of all positive integers. Such sequences are also called two resp. one sided infinite binary words. Let Pa(n) be the number of different binary words of length n occurring in a. The mapping n 7→ Pa(n) is called the complexity function of a. It is easily seen that the complexity function is bounded if and only if a is (in the one sided case: eventually) periodic. Among all aperiodic sequences Sturmian sequences have minimal complexity, namely P (n) = n+ 1. This is the combinatorial approach to characterise Sturmian sequences, which has been introduced in [6] and [7]. A different characterisation uses the symbolic coding of irrational rotations. If T = R/Z denotes the circle group (one dimensional torus) and M is a segment of T of angle 2πα with irrational α ∈ T, then the definition ak = 1 if and only if kα ∈M defines a Sturmian sequence (cf. for instance [2]). For understanding the definition of a Hartman sequence the second approach is more appropriate. Replace T by more general compact abelian groups C with normalised Haar measure μ = μC and replace α by any ergodic group translation. This means that we are interested in the transformation T : C 7→ C, T : x 7→ x + g, where g is a generating element of C, i.e. the orbit {kg : k ∈ Z} is dense in C. Thus C is required to be monothetic. C can also be interpreted as a group compactification of Z since C is the closure of the image of Z under the dense homomorphic embedding ι : Z → C, ι(k) = kg. (Note that for group compactifications one usually does not require ι to be injective. But to avoid trivial case distinctions we will demand that ι(Z) is infinite.) This approach is particularly appropriate for Theorems 1 and 2. For the classical theory of ergodic group translations we refer for instance to [11]. A set M ⊂ C is called a (μC-)continuity set if μC(∂M) = 0 holds for its topological boundary ∂M . For a continuity set M consider the induced binary sequence a = (ak)k=−∞ defined by ak = 1 if T (0C) ∈M and ak = 0 otherwise. Such sequences are called Hartman sequences. The set H ⊂ Z defined by k ∈ H if and only if ak = 1 is accordingly called a Hartman set. Thus, by definition, a Hartman set H is the preimage H = ι−1(M) of a continuity set M ⊆ C where (C, ι) is a group compactification of Z and we can write a = 1H . Note that for C = T and g = (α1, . . . , αs), where the family {1, α1, . . . , αs} is linearly independent over Z, Hartman sequences are binary coding sequences of Kronecker sequences. As a consequence of uniform distribution of ergodic group translations (for the theory of uniform distribution of sequences we refer to [9]), Hartman sequences share some nice properties with Sturmian sequences (cf. [5], [10]). In particular each finite subword occurs with a uniform asymptotic density.