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

TL;DR

This paper develops new explicit constructions of large subsets in modular integer vector spaces that avoid arithmetic progressions, providing improved lower bounds for their sizes across various parameters.

Contribution

The authors introduce novel explicit methods to construct progression-free sets in \\mathbb{Z}_m^n, improving known lower bounds for their maximal sizes for different progression lengths and conditions.

Findings

Established new lower bounds for r_k(\\mathbb{Z}_m^n) with explicit constructions.

Derived bounds depend on prime factors and congruence conditions of m.

Provided improved bounds for prime moduli p ≤ 31 and progression lengths 4 to 8.

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 $k\geq 5$ is odd and $P^{-}(m)\geq (k+2)/2$, 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 $k\geq 4$ is even, $P^{-}(m) \geq k$ and $m \equiv -1 \bmod k$, 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 $p \leq 31$ and progression lengths $4 \leq k \leq 8$.