For an integer $$q\ge 2$$ q ≥ 2 , a q-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 every q-recursive sequence is q-regular in the sense of Allouche and Shallit and that a q-linear representation of the sequence can be computed easily by using the coefficients from the recurrence relations. Detailed asymptotic results for q-recursive sequences are then obtained based on a general result on the asymptotic analysis of q-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.

No 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 . If $$k\ge 4$$ k ≥ 4 is even, $$P^{-}(m) \ge k$$ P - ( m ) ≥ k and $$m \equiv -1 \bmod k$$ m ≡ - 1 mod k , then $$\begin{aligned} r_{k}(\mathbb {Z}_{m}^{n}) \gg _{m,k} \frac{\bigl \lfloor \frac{k-2}{k}m + 2\bigr \rfloor ^{n}}{n^{\lfloor \frac{k-2}{k}m + 1\rfloor /2}}. \end{aligned}$$ r k ( Z m n ) ≫ m , k ⌊ k - 2 k m + 2 ⌋ n n ⌊ k - 2 k m + 1 ⌋ / 2 . Moreover, we give some further improved lower bounds on $$r_k(\mathbb {Z}_p^n)$$ r k ( Z p n ) for primes $$p \le 31$$ p ≤ 31 and progression lengths $$4 \le k \le 8$$ 4 ≤ k ≤ 8 .

In spite of a recent breakthrough on upper bounds of the size of cap sets (by Croot, Lev and Pach and by Ellenberg and Gijswijt), the classical cap set constructions had not been affected. In this work, we introduce a very different method of construction for caps in all affine spaces with odd prime modulus p. Moreover, we show that for all primes p≡50.28emmod0.28em6$p \equiv 5 \bmod 6$ with p⩽41$p \leqslant 41$, the new construction leads to an exponentially larger growth of the affine and projective caps in AGfalse(n,pfalse)${\rm AG}(n,p)$ and PGfalse(n,pfalse)${\rm PG}(n,p)$. For example, when p=23$p=23$, the existence of caps with growth (8.0875…)n$(8.0875\ldots )^n$ follows from a three‐dimensional example of Bose, and the only improvement had been to (8.0901…)n$(8.0901\ldots )^n$ by Edel, based on a six‐dimensional example. We improve this lower bound to (9−ofalse(1false))n$(9-o(1))^n$.