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Erdős-Deep Families of Arithmetic Progressions

Let $A \subseteq \Z_n$ with $|A| = k$ for some $k \in \Z^+$. We consider the metric space $(\Z_n,\delta)$ in which $\delta$ is the distance metric on $\Z_n$ defined as follows: for every $x,y \in \Z_n$, $\delta(x,y) = |x-y|_n$ where $|z|_n = \min(z,n-z)$ for $z \in \{0,\ldots,n-1\}$. We say that $A$ is \emph{Erd\H{o}s-deep} if, for every $i \in \{1,2,\dots,k-1\}$, there is a positive number $d_i$ satisfying
$$|\{\{x,y\} \subseteq A: \delta(x,y)=d_i\}|=i.$$

Erd\H{o}s-deep sets in $\Z_n$ have been previously classified as translates of: $\{0,1,2,4\}$ when $n = 6$; and, modular arithmetic progressions $\{0,g,2g,\cdots,(k-1)g\} \subseteq \Z_n$ for some generator $g$ and size $k$.

Erd\H{o}s-deep sets have primarily been considered in metric spaces $(\Z_n,\delta)$ and $(\R^d,\norm{\cdot})$ for $d = 2$, but some exploration for $d > 2$ has been done as well.

We introduce the notion of an \emph{Erd\H{o}s-deep family}. Let $\mathcal{F}=\{A_1,A_2,\dots,A_s\}$, where $A_1,\ldots, A_s \subseteq \Z_n$. Then we say $\mathcal{F}$ is Erd\H{o}s-deep if for some $k \in \Z^+$, for every $i \in \{1,2,\dots,k-1\}$ there is exactly one positive number $d_i$ satisfying
$$\sum_{j=1}^s |\{\{x,y\} \subseteq A_j: \delta(x,y)=d_i\}|=i,$$
and no such $d_i$ for any $i \ge k$.

We provide a complete existence theorem for Erd\H{o}s-deep pairs of arithmetic progressions $A_1,A_2 \subseteq \Z_n$ and also give a conjectured classification for Erd\H{o}s-deep families of three arithmetic progressions. Using an identity on triangular numbers, we show a general construction for larger families whose size $s$ is the square of an integer. This construction suggests the existence of Erd\H{o}s-deep families often relies on such number-theoretic identities.

We define an extremal case of the Erd\H{o}s-deep family in $(\Z_n,\delta)$ in which both the distances and multiplicities are in $\{1,\ldots,k-1\}$; such families are called Winograd families. We conjecture that Winograd families of arithmetic progressions do not exist in the metric space $(\Z,|\cdot|)$.

Erd\H{o}s-deep sets in $(\Z_n,\delta)$ correspond to a class of interesting musical rhythms. We conclude this work with a variety of musical demonstrations and original compositions using Erd\H{o}s-deep rhythm families as a creative constraint in composing multi-voiced rhythms. / Graduate
Date30 August 2022
CreatorsGaede, Tao
ContributorsDukes, Peter
Source SetsUniversity of Victoria
LanguageEnglish, English
Detected LanguageEnglish
RightsAvailable to the World Wide Web

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