论文标题
具有距离倍数的间隔的模块化算术进程家族
Families of modular arithmetic progressions with an interval of distance multiplicities
论文作者
论文摘要
给定一个家庭$ \ MATHCAL {f} = \ {a_1，\ dots，a_s \} $ $ \ Mathbb {z} _n $的子集，定义$Δ\ Mathcal {f} $是所有（cyclic）差异$（cyclic）差异$（y） $ x \ neq y $，对于某些$ i = 1，\ dots，s $。从Erds的Euclidean距离问题中汲取灵感，我们说$ \ Mathcal {f} $如果在$δ\ Mathcal {f}中发生的多个距离{f} $确切地为$ 1,2，\ dots，k-1 $ k-1 $ k-1 n Integer $ k $ k-$ k-$。在$ s = 1 $的情况下，众所周知，$ \ mathbb {z} _n $中的模块化算术进展（在温和条件下）；相反，除了$ n = 6 $时，AP是唯一的集合，除了一个零星的情况。在这里，我们详细考虑了$ s = 2 $的情况。特别是，当每个$ a_i $都是$ \ mathbb {z} _n $中的算术进程时，我们将Erdős-Deep对$ \ {a_1，a_2 \} $分类。当$ s $是正方形的整数时，我们还为一类较宽的Erdős-Deep家族$ \ {a_1，\ dots，a_s \} $构造了一个更广泛的阶层。
Given a family $\mathcal{F}=\{A_1,\dots,A_s\}$ of subsets of $\mathbb{Z}_n$, define $Δ\mathcal{F}$ to be the multiset of all (cyclic) distances dist$(x,y)$, where $\{x,y\} \subset A_i$, $x \neq y$, for some $i=1,\dots,s$. Taking inspiration from a Euclidean distance problem of Erdős, we say that $\mathcal{F}$ is Erdős-deep if the multiplicities of distances that occur in $Δ\mathcal{F}$ are precisely $1,2,\dots,k-1$ for some integer $k$. In the case $s=1$, it is known that a modular arithmetic progression in $\mathbb{Z}_n$ achieves this property (under mild conditions); conversely, APs are the only such sets, except for one sporadic case when $n=6$. Here, we consider in detail the case $s=2$. In particular, we classify Erdős-deep pairs $\{A_1,A_2\}$ when each $A_i$ is an arithmetic progression in $\mathbb{Z}_n$. We also give a construction of a much wider class of Erdős-deep families $\{A_1,\dots,A_s\}$ when $s$ is a square integer.
