Le masque de Steinhaus : (1958)

On cherche un ensemble M de points du plan qui aurait l'étonnant pouvoir suivant :

Quelque soit le déplacement du masque, (translation, rotation) que l'on effectue, Une et une seule lumière est visible.

l'2001i:52022

Adhikari, Sukumar Das(6-HCRI-MP)

A problem of Steinhaus concerning the existence of a plane set with a certain property. Number theory and its applications (Kyoto, 1997), 1--5, Dev. Math., 2, Kluwer Acad. Publ., Dordrecht, 1999.

52C05 (11H06)

In the late 1950s, Steinhaus asked if there exists a planar set which, however translated and rotated, always contains exactly

one integer lattice point. The answer to this question is still not known. The paper under review is a nice exposition of the partial

results on this Steinhaus problem until about 1997. It is very well written and presents nicely the important ideas in their time

frame.

In my opinion two remarks should be emphasized. The first is that the problem becomes essentially different if one requires that

the set be measurable, allowing for the use of tools such as harmonic analysis and measure theory in disproving the existence

of Steinhaus sets under extra assumptions. By contrast, searching for any not necessarily measurable set allows one to use

tools of set theory in the constructions. The second remark is another way of phrasing the Steinhaus problem: A set $E$ is a

Steinhaus set if and only if any rotation of the set $E$ translated at the locations $ Z^2$ forms a tiling of the plane.

{For the entire collection see MR 2000j:11005.}

91i:11072

Beck, József(H-EOTVO-C)

On a lattice-point problem of H. Steinhaus.

Studia Sci. Math. Hungar. 24 (1989), no. 2-3, 263--268.

11H16

References: 0

Reference Citations: 2

Review Citations: 3

For the set $S\subset\bold R^2$ containing the origin $\theta$ denote $S(\tau,x):=\{\tau(y)+x\colon y\in S\}$, $x\in\bold

R^2$, where $\tau\in[0,2\pi)$ and $\tau(y)$ means the rotation of $y$ by the angle $\tau$. H. Steinhaus asked whether there

exists $S$ such that $S(\tau,x)$ contains exactly one point with integer coordinates (lattice point) for all $\tau$ and $x$. (The

latter property of $S$ is called the Steinhaus property.) W. Sierpi\'nski \ref[Fund. Math. 46 (1959), 191--194; MR 21 #851]

showed that an $S$ having the Steinhaus property can be neither compact nor bounded open. In the paper it is proved that

such an $S$ cannot be bounded and Lebesgue measurable. Sierpi\'nski's remark can be proved quite easily using only

translations of $S$, i.e., $\tau=0$ (this is demonstrated in the paper as well). The set $S:=[0,1)^2$ trivially has the Steinhaus

property for all translations, which shows that to prove the statement one has to use $\tau\not=0$. The author proves his

statement using some facts on Fourier transforms. The proof is based on a lemma that seems to be interesting in itself. The

lemma says that there is $\tau\in[0,2\pi)$ and a nonzero lattice point $u$ such that the value of the Fourier transform of the

characteristic function of the set $S(\tau,\theta)$ is nonzero at the point $2\pi u$.

Reviewed by Béla Uhrin