On Arithmetic Index in the Generalized Thue-Morse Word

On long arithmetic progressions in binary Morse-like words

“…Other potential generalisations include the investigation of other substitution sequences, either with more letters (see [24] for results on a different class of generalised Thue-Morse sequences) or for non-bijective substitutions, or quantitative versions of normality results [4,26,27,21].…”

“…Therefore, v i+2d = 0 and A(d) 3. Parshina proved the following result [23], as well as a generalisation to similar sequences in larger alphabets [24,25]. Her proofs for the Thue-Morse case [23] are based on a detailed analysis of binary arithmetic.…”

“…In this paper, we will give exact expressions of A(d) for certain values of d, first for the Thue-Morse sequence, and later also for generalised Thue-Morse sequences, which are different generalisations from those considered in [23,24,25]. In particular, all our sequences are on a two-letter alphabet.…”

“…Also, it is clear from the substitution rule (3) that the language of θ p,q is (Q+1)-powerfree (in fact, (Q + ε)-powerfree for any ε > 0), generalising the cube-freeness (overlap-freeness) of the Thue-Morse case. We note that Parshina also considered generalised Thue-Morse words [24], but in her work the generalisation is to larger alphabets. Here, we consider a generalisation of the Thue-Morse sequence along the lines of [15,7,6], restricting ourselves to the binary case.…”

“…Our results generalise to a set of binary bijective substitutions which are analysed in Section 3. We note that our generalised Thue-Morse substitutions [15,19,7,6] are different from those considered by Parshina [24]. In Section 4, we establish a bound for the existence of certain arithmetic progressions for general bijective substitutions on an arbitrary alphabet, before concluding in Section 5 with some open questions and proposed directions for further research.…”