Tiling by translates of a function: results and open problems

Tiling by translates of a function: results and open problems, Discrete Analysis 2021:12, 24 pp.

Let $f$ be a function from $\mathbb R$ to $\mathbb R$, and for each $\lambda\in\mathbb R$, let $T_\lambda$ be the translate of $f$ by $\lambda$, that is, the function given by the formula
$$T_\lambda f(x) = f(x-\lambda).$$
We say that $f$ tiles $\mathbb R$ at level $w$ if there is a set $\Lambda\subset\mathbb R$ such that the function $\sum_{\lambda\in\Lambda}T_\lambda f$ is equal (or equal almost everywhere) to the constant function that takes value $w$. (The convergence in the sum is required to be absolute.) The word “tiles” is used because if $f$ is the characteristic function of some set $S$ and $w=1$, then a tiling of $\mathbb R$ by $f$ is (up to sets of measure zero) a partition of $\mathbb R$ into translates of $S$.

This article is concerned with the general problem of understanding what such tilings can look like, with a particular emphasis on what the sets $\Lambda$ can look like. One can of course consider tilings of other groups such as $\mathbb R^d$ with $d>1$, but the one-dimensional case turns out to be particularly interesting. (There are also important open problems such as Fuglede’s conjecture that concern tilings of finite groups.) The paper provides both new results and a useful survey of the area – useful because the existing results were scattered amongst several papers and not always easy to find.

Since one can always multiply $f$ by a scalar, the precise value of $w$ is not important – what matters is merely whether it is zero or non-zero. A simple example of a tiling at level 1 is to take $f$ to be the characteristic function of the interval $[0,1)$ and $\Lambda$ to be $\mathbb Z$. A slightly more complicated one is to take $f$ to be the characteristic function of $[0,1)\cup [2,3)$ and to take $\Lambda$ to be $4\mathbb Z\cup(4\mathbb Z+1)$.

Note that the value taken by the function $\sum_{\lambda\in\Lambda}T_\lambda f$ at $x$ is $\sum_{\lambda\in\Lambda}f(x-\lambda)$, which we can write as $\sum_{y+z=x}f(y)1_\Lambda(z)$. That is, we have a kind of convolution of $f$ with the characteristic function of $\Lambda$. This thought leads naturally to the observation that if we have $n$ disjoint arithmetic progressions $a_i\mathbb Z+b_i$, then there is a compactly supported function $f$ such that $\sum_{\lambda\in\Lambda}T_\lambda f$ tiles $\mathbb R$, where $\Lambda$ is the union of the arithmetic progressions. Indeed, we can take
$$f=1_{[0,a_1)}*1_{[0,a_2)}*\dots*1_{[0,a_n)}.$$
Writing $\Lambda_i$ for the progression $a_i\mathbb Z+b_i$, we see that $f*\Lambda_i$ is equal to the convolution that defines $f$, but with $1_{[0,a_i)}$ replaced by the constant function 1. Thus, $f*\Lambda_i$ is itself a constant function, and summing over $i$ we again obtain a constant function.

This one can think of as the basic class of examples of translation sets $\Lambda$ for tilings of $\mathbb R$. (It is tempting to try to switch round the roles of $f$ and $\Lambda$, letting $f$ be a sum of two functions that tile $\mathbb R$ using sets $\Lambda_1$ and $\Lambda_2$ and letting $\Lambda=\Lambda_1*\Lambda_2$ for some suitable definition of the convolution. Unfortunately this doesn’t work, as the set of numbers $\lambda_1+\lambda_2$ is typically much too dense.)

The interesting questions begin as soon as one moves beyond this class. Indeed, virtually every sensible question one might wish to ask turns out to be a non-trivial problem. Must $\Lambda$ have a “uniform density” in the sense that there is some $\delta$ such that $|\Lambda\cap[x,x+r]|=(\delta+o(1))r$ for all $x$ (where the convergence implicit in the $o(1)$ notation is uniform in $x$)? Does the integral of $f$ have to be zero if $w=0$ and non-zero if $w\ne 0$? If $f\in L_1(\mathbb R)$ and $w=1$, it is easy to see that $\Lambda$ cannot have density zero, but what if $w=0$? These (to which the answers are known) and many other questions are discussed in the paper. A particular highlight of the results proved is the construction of a tiling of $\mathbb R$ for any given level $w$ such that $\Lambda$ is a set of unbounded density (meaning that for every $C$ there is an interval $[a,b]$ such that $|\Lambda\cap[a,b]|\geq C(b-a)$). The function $f$ in this case belongs to the Schwartz class.

The paper concludes with some very nice open problems. To mention just one, it is not known whether there is a tiling of $\mathbb R$ such that $\Lambda$ has bounded density but not uniform density. Such a tiling does not exist if $w\ne 0$ (this is a result of Kolountzakis and Lagarias from 1996) but the question remains open if $w=0$.