Large Subsets of ${\mathbb{Z}}_m^n$ without Arithmetic Progressions

Abstract

For integers $m$ and $n$, we study the problem of finding good lower bounds for the size of progression-free sets in $({\mathbb{Z}}_m^n,+)$. Let $r_k({\mathbb{Z}}_m^n)$ denote the maximal size of a subset of ${\mathbb{Z}}_m^n$ without arithmetic progressions of length $k$ and let $P^{-}(m)$ denote the least prime factor of $m$. We construct explicit progression-free sets and obtain the following improved lower bounds for $r_k({\mathbb{Z}}_m^n)$: If $k\ge 5$ is odd and $P^{-}(m)\ge (k+2)/2$, then \[r_k(\mathbb{Z}_m^n) \gg_{m,k} \frac{\bigl\lfloor \frac{k-1}{k+1}m +1\bigr\rfloor^{n}}{n^{\lfloor \frac{k-1}{k+1}m \rfloor/2}}. \] If $k\ge 4$ is even, $P^{-}(m)\ge k$ and $m\equiv -1\mathrm{mod}k$, then \[r_{k}(\mathbb{Z}_{m}^{n}) \gg_{m,k} \frac{\bigl\lfloor \frac{k-2}{k}m + 2\bigr\rfloor^{n}}{n^{\lfloor \frac{k-2}{k}m + 1\rfloor/2}}.\] Moreover, we give some further improved lower bounds on $r_k({\mathbb{Z}}_p^n)$ for primes $p\le 31$ and progression lengths $4\le k\le 8$.

10 pages

References in corpus (1)

• Exponentially Larger Affine and Projective Caps

Cited by in corpus (1)

• Maximal line-free sets in ${\mathbb{F}}_p^n$