Families of Modular Arithmetic Progressions with an Interval of Distance Multiplicities

Given a family $\mathcal{F}=\{A_1,\dots,A_s\}$ of subsets of $\Z_n$, define $\Delta \cF$ to be the multiset of all (cyclic) distances dist$(x,y)$, where $\{x,y\} \subset A_i$, $x \neq y$, for some $i \in \{1,\dots,s\}$. Taking inspiration from a Euclidean distance problem of Erd\H{o}s, we say that $\mathcal{F}$ is \emph{Erd\H{o}s-deep} if the multiplicities of distances that occur in $\Delta \cF$ 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 $\Z_n$ achieves this property (under mild conditions); conversely, arithmetic progressions 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 Erd\H{o}s-deep pairs $\{A_1,A_2\}$ when each $A_i$ is an arithmetic progression in $\Z_n$. We also give a construction of a much wider class of Erd\H{o}s-deep families $\{A_1,\dots,A_s\}$ when $s$ is a square integer.

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