论文标题
$ \ mathbb {z} _m^n $的大子集无算术进度
Large Subsets of $\mathbb{Z}_m^n$ without Arithmetic Progressions
论文作者
论文摘要
对于整数$ m $和$ n $，我们研究了在$（\ Mathbb {z} _ {m}^{n}，+）$中找到无进展集的良好下限的问题。令$ r_ {k}（\ mathbb {z} _ {m}^{n}）$表示$ \ mathbb {z} _ {z} _ {m}^{n} $的最大大小，而无需$ k $的arithmetic进度$ p^{ - $ p^{ - } $ m。我们构建明确的无进展集，并获得以下改进的下限，以$ r_ {k}（\ m athbb {z} _ {m}^{n}）$：如果$ k \ geq 5 $是奇数且$ p^{ - } - } - }（ - }（m）\ geq（k+2）/2 $，然后\ gg_ {m，k} \ frac {\ bigl \ lfloor \ frac {k-1} {k+1} m +1 \ bigr \ rfloor^{n}}} {n^{ \]如果$ k \ geq 4 $均为偶数，$ p^{ - }（m）\ geq k $和$ m \ equiv -1 \ bmod k $，则\ [r_ {k {k {k Mathbb {z}} \ frac {k-2} {k} m + 2 \ bigr \ rfloor^{n}} {n^{\ lfloor \ frac {k-2} {k-2} {k} m + 1 \ rfloor/2}}。 Primes $ p \ leq 31 $和进度长度$ 4 \ leq k \ leq 8 $。
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 $4 \leq k \leq 8$.
