On sets with missing differences in compact abelian groups

TL;DR

This paper investigates the Motzkin density problem in compact abelian groups, establishing conditions under which maximal density avoiding certain difference sets is achieved by periodic sets, with explicit formulas in some cases.

Contribution

It extends the Motzkin density problem to compact abelian groups using ergodic theory and provides new results on periodicity and explicit formulas for specific cases.

Findings

Motzkin density can be rational for up to three missing differences.

Periodic sets of maximal density exist in certain cases, including rank 1 and rank r-1 lattices.

Counterexamples show periodicity does not always hold, as per Greenfeld--Tao.

Abstract

A much-studied problem posed by Motzkin asks to determine, given a finite set $D$ of integers, the so-called Motzkin density for $D$, i.e., the supremum of upper densities of sets of integers whose difference set avoids $D$. We study the natural analogue of this problem in compact abelian groups. Using ergodic-theoretic tools, this is shown to be equivalent to the following discrete problem: given a lattice $\Lambda\subset \mathbb{Z}^r$, letting $D$ be the image in $\mathbb{Z}^r/\Lambda$ of the standard basis, determine the Motzkin density for $D$ in $\mathbb{Z}^r/\Lambda$. We study in particular the periodicity question: is there a periodic $D$-avoiding set of maximal density in $\mathbb{Z}^r/\Lambda$? The Greenfeld--Tao counterexample to the periodic tiling conjecture implies that the answer can be negative. On the other hand, we prove that the answer is positive in several cases, including the case rank$(\Lambda)=1$ (in which we give a formula for the Motzkin density), the case rank$(\Lambda)=r-1$, and hence also the case $r\leq 3$. It follows that, for up to three missing differences, the Motzkin density in a compact abelian group is always a rational number.