Large subsets of $$\mathbb {Z}_m^n$$ without arithmetic progressions

Abstract

AbstractFor 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 .