Tiling With Punctured Intervals

Abstract

It was shown by Gruslys, Leader and Tan that any finite subset of Z n tiles Z d for some d. The first non-trivial case is the punctured interval, which consists of the interval {−k, . . . , k} ⊂ Z with its middle point removed: they showed that this tiles Z d for d = 2k 2 , and they asked if the dimension needed tends to infinity with k. In this note we answer this question: we show that, perhaps surprisingly, every punctured interval tiles Z 4 .

Punctured Intervals Tile Z3

“…This theorem answers two concrete questions posed by Metrebian [4,Question 10,11]. As a corollary, the least d for which…”

“…This theorem answers two concrete questions posed by Metrebian [4, Question 10,11]. As a corollary, the least d for which tiles equals , answering [2, Question 21].…”

“…As a warm up and for completeness of presentation, we construct three partial tilings of the plane satisfying the conditions of Lemma 2.1 when T is the symmetric punctured interval with . This was also proven in [4, Theorem 3]. Lemma If , then tiles .…”

“…In this section, we will prove that finding certain partial tilings is enough to conclude that a whole tiling does exist. This is done in Lemma 2.1 which is a generalization of [4, Lemma 4]. Lemma Let T be the one‐dimensional tile .…”

“…This is done in Lemma 3.2. Metrebian [4] did have such examples already when . An important ingredient to prove the validity of this infinite family of partial tilings is some elementary number theory.…”