In this note, we precisely elaborate the connection between recognisable series (in the sense of Berstel and Reutenauer) and q-regular sequences (in the sense of Allouche and Shallit) via their linear representations. In particular, we show that the minimisation algorithm for recognisable series can also be used to minimise linear representations of q-regular sequences.

Abstract For integers m and n, we study the problem of finding good lower bounds for the size of progression-free sets in $$(\mathbb {Z}_{m}^{n},+)$$ ( Z m n , + ) . Let $$r_{k}(\mathbb {Z}_{m}^{n})$$ r k ( Z m n ) denote the maximal size of a subset of $$\mathbb {Z}_{m}^{n}$$ Z m n without arithmetic progressions of length k and let $$P^{-}(m)$$ P - ( m ) denote the least prime factor of m. We construct explicit progression-free sets and obtain the following improved lower bounds for $$r_{k}(\mathbb {Z}_{m}^{n})$$ r k ( Z m n ) : If $$k\ge 5$$ k ≥ 5 is odd and $$P^{-}(m)\ge (k+2)/2$$ P - ( m ) ≥ ( k + 2 ) / 2 , then $$\begin{aligned} r_k(\mathbb {Z}_m^n) \gg _{m,k} \frac{\bigl \lfloor \frac{k-1}{k+1}m +1\bigr \rfloor ^{n}}{n^{\lfloor \frac{k-1}{k+1}m \rfloor /2}}. \end{aligned}$$ r k ( Z m n ) ≫ m , k ⌊ k - 1 k + 1 m + 1 ⌋ n n ⌊ k - 1 k + 1 m ⌋ / 2 .

Abstract For an integer $$q\ge 2$$ q≥2 , aq-recursive sequence is defined by recurrence relations on subsequences of indices modulo some powers of q. In this article,q-recursive sequences are studied and the asymptotic behavior of their summatory functions is analyzed. It is shown that everyq-recursive sequence isq-regular in the sense of Allouche and Shallit and that aq-linear representation of the sequence can be computed easily by using the coefficients from the recurrence relations. Detailed asymptotic results forq-recursive sequences are then obtained based on a general result on the asymptotic analysis ofq-regular sequences. Three particular sequences are studied in detail: We discuss the asymptotic behavior of the summatory functions of Stern’s diatomic sequence, the number of non-zero elements in some generalized Pascal’s triangle and the number of unbordered factors in the Thue–Morse sequence. For the first two sequences, our analysis even leads to precise formulæ without error terms.