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On sets with missing differences in compact abelian groups

Pablo Candela, Fernando Chamizo, Antonio Córdoba

TL;DR

The paper analyzes Motzkin density Md_G(D) for finite missing-difference sets in compact abelian groups, reducing the problem to a discrete Motzkin density Md_{\mathbb{Z}^r/\Lambda}(B_\Lambda) determined by a lattice Λ. It provides an explicit formula in the rank-1 case, Md_{\mathbb{Z}^r/\Lambda}(B_\Lambda)=\lfloor\|m\|_{\ell^1}/2\rfloor/\|m\|_{\ell^1}, and proves periodic attainment for rank(\Lambda)=r-1, with consequences that Md is rational for |D|≤3 when r≤3. The work also establishes general reductions to Z^k and develops lower and upper bounds in the circle group via κ_{\mathbb{T}}(D) and Fourier-analytic methods, illustrating connections to tiling, packing, and the Greenfeld–Tao counterexample. Overall, it clarifies when maximal D-avoiding sets can be chosen periodically and when densities are rational, linking continuous-group problems to discrete lattice geometry and spectral techniques.

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 $Λ\subset \mathbb{Z}^r$, letting $D$ be the image in $\mathbb{Z}^r/Λ$ of the standard basis, determine the Motzkin density for $D$ in $\mathbb{Z}^r/Λ$. We study in particular the periodicity question: is there a periodic $D$-avoiding set of maximal density in $\mathbb{Z}^r/Λ$? 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$(Λ)=1$ (in which we give a formula for the Motzkin density), the case rank$(Λ)=r-1$, 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.

On sets with missing differences in compact abelian groups

TL;DR

The paper analyzes Motzkin density Md_G(D) for finite missing-difference sets in compact abelian groups, reducing the problem to a discrete Motzkin density Md_{\mathbb{Z}^r/\Lambda}(B_\Lambda) determined by a lattice Λ. It provides an explicit formula in the rank-1 case, Md_{\mathbb{Z}^r/\Lambda}(B_\Lambda)=\lfloor\|m\|_{\ell^1}/2\rfloor/\|m\|_{\ell^1}, and proves periodic attainment for rank(\Lambda)=r-1, with consequences that Md is rational for |D|≤3 when r≤3. The work also establishes general reductions to Z^k and develops lower and upper bounds in the circle group via κ_{\mathbb{T}}(D) and Fourier-analytic methods, illustrating connections to tiling, packing, and the Greenfeld–Tao counterexample. Overall, it clarifies when maximal D-avoiding sets can be chosen periodically and when densities are rational, linking continuous-group problems to discrete lattice geometry and spectral techniques.

Abstract

A much-studied problem posed by Motzkin asks to determine, given a finite set of integers, the so-called Motzkin density for , i.e., the supremum of upper densities of sets of integers whose difference set avoids . 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 , letting be the image in of the standard basis, determine the Motzkin density for in . We study in particular the periodicity question: is there a periodic -avoiding set of maximal density in ? 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 (in which we give a formula for the Motzkin density), the case rank, and hence also the case . It follows that, for up to three missing differences, the Motzkin density in a compact abelian group is always a rational number.
Paper Structure (8 sections, 27 theorems, 39 equations)

This paper contains 8 sections, 27 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Initial remarks on Problem \ref{['prob:CAGMotzkin']}, and statements of main results
  3. On Motzkin densities in $\mathbb{Z}^k$
  4. Connections with tiling and packing problems
  5. Motzkin densities in the case rank$(\Lambda)=1$
  6. The case rank$(\Lambda)=r-1$
  7. Bounds for the Motzkin problem in $\mathbb{T}$
  8. Reduction to metrizable compact abelian groups

Key Result

Theorem 1.2

For any compact abelian group $\mathop{\mathrm{G}}\nolimits$ and any set $D\subset \mathop{\mathrm{G}}\nolimits\setminus\{0\}$ with $|D|\leq 3$, we have $\mathop{\mathrm{Md}}\nolimits_{\mathop{\mathrm{G}}\nolimits}(D)\in \mathbb{Q}$.

Theorems & Definitions (59)

  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2: See Lemma 2.8 in CCRS
  • Lemma 2.3
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Remark 2.7
  • Proposition 2.11
  • ...and 49 more