Gowers norms for automatic sequences

Abstract

We show that any automatic sequence can be separated into a structured part and a Gowers uniform part in a way that is considerably more efficient than guaranteed by the Arithmetic Regularity Lemma. For sequences produced by strongly connected and prolongable automata, the structured part is rationally almost periodic, while for general sequences the description is marginally more complicated. In particular, we show that all automatic sequences orthogonal to periodic sequences are Gowers uniform. As an application, we obtain for any l ≥ 2 and any automatic set A ⊂ N0 lower bounds on the number of l-term arithmetic progressions -contained in A -with a given difference. The analogous result is false for general subsets of N0 and progressions of length ≥ 5.

(Logarithmic) densities for automatic sequences along primes and squares

“…This proposition will allow us to approximate an automatic sequence (a(n)) n∈N by the primitive automatic sequences (b i (n)) n∈N . 5 We start by proving an auxiliary result, which shows that the M i are k-automatic sets, i.e. the indicator function is k-automatic.…”

“…In particular, we can choose L ν = g(k λν ), as g(k λ(ν+1) )/g(k λν ) → k λβ for any λ ∈ N ≥1 . Moreover, Lemma 2.1 shows that we can replace log(g(k λν )) by log(k βλν ) in (5).…”

Automata and finite order elements in the Nottingham group

“…and with t = x 2 y −1 we compute that σ(t) = (x 2 + 1)/(y + x 2 + 1). By eliminating x and y from these last two equations and the two equations in (18), we find that σ = σ(t) satisfies the following (irreducible) equation over F 2 (t): Considering the sum and product of the three solutions, we find that there is a unique solution with σ = t + O(t 2 ). The corresponding automaton with initial coefficients t + t 2 + t 3 + t 4 + t 6 + O(t 7 ) and Equation (19) produced by the algorithm is displayed in Figure 4.…”

“…Finally, in Section 13 we briefly discuss the synchronisation properties of some of our automata, in relation to a 'structured/random' decomposition of automatic sequences in [18].…”

“…The automata corresponding to σ J and σ •3 J have two absorbing states, and every state has a path to one of these two states; this is enough to conclude that these automata are forwards synchronising (cf. [18,Lemma 3.2]). Finally, the automata for σ •2 CS and σ CS have two subgraphs that are backwards synchronising and the start vertex is connected by an outgoing edge to these two subgraphs; it follows that the value a n of the corresponding sequence depends on the value of a backwards synchronising automaton (the product automaton for the subgraphs) and on the value of the sequence (n mod 2), which is forwards synchronising.…”