On sets with missing differences in compact abelian groups
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 er...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión enviada para evaluación y publicación |
| Fecha de publicación: | 2025 |
| País: | España |
| Institución: | Consejo Superior de Investigaciones Científicas (CSIC) |
| Repositorio: | DIGITAL.CSIC. Repositorio Institucional del CSIC |
| OAI Identifier: | oai:digital.csic.es:10261/425184 |
| Acceso en línea: | http://hdl.handle.net/10261/425184 https://www.scopus.com/inward/record.uri?eid=2-s2.0-105025658668&doi=10.1007%2Fs10474-025-01574-8&partnerID=40&md5=f311397e8ae55f56567542cc37b652b1 |
| Access Level: | acceso abierto |
| Palabra clave: | missing difference Motzkin density Motzkin problem tiling set |
| Sumario: | 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⊂Zr, letting D be the image in Zr/Λ of the standard basis, determine the Motzkin density for D in Zr/Λ. We study in particular the periodicity question: is there a periodic D-avoiding set of maximal density in Zr/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≤3. It follows that, for up to three missing differences, the Motzkin density in a compact abelian group is always a rational number. © The Author(s), under exclusive licence to Akadémiai Kiadó Zrt 2025. |
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