Edge-Isoperimetric Problem for Cayley Graphs and Generalized Takagi Functions

Abstract

Let $G$ be a finite abelian group of exponent $m\ge 2$. For subsets $A,S\subseteq G$, denote by $\partial_S(A)$ the number of edges from $A$ to its complement $G\setminus A$ in the directed Cayley graph, induced by $S$ on $G$. We show that if $S$ generates $G$, and $A$ is nonempty, then $\partial_S(A) \ge \frac{e}m\,|A|\ln\frac{|G|}{|A|}\, . $ Here the coefficient $e=2.718\ldots$ is best possible and cannot be replaced with a number larger than $e$. For homocyclic groups $G$ of exponent $m$, we find an explicit closed-form expression for $\partial_S(A)$ in the case where $S$ is the “standard" generating subset of $G$, and $A$ is an initial segment of $G$ with respect to the lexicographic order induced by $S$. Namely, we show that in this situation $ \partial_S(A) = |G|\,\omega_m(|A|/|G|), $ where $\omega_2$ is the Takagi function, and $\omega_m$ for $m\ge3$ is an appropriate generalization thereof. This particular case is of special interest, since for $m\in\{2,3,4\}$ it is known to yield the smallest possible value of $\partial_S(A)$, over all sets $A\subseteq G$ of given size. We give this classical result a new proof, somewhat different from the standard one. We also give a new, short proof of the Boros--Páles inequality $\omega_2(\frac{x+y}2) \le \frac{\omega_2(x)+\omega_2(y)}2 + \frac12\,|y-x|,$ establish an extremal characterization of the Takagi function as the (pointwise) maximal function, satisfying this inequality and the boundary condition $\max\{\omega_2(0),$ $\omega_2(1)\}\le 0$, and obtain similar results for the $3$-adic analogue $\omega_3$ of the Takagi function.