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A remark on sets having the Steinhaus property

1996
*
Combinatorica
*

*A*point

*set*satisfies

*the*

*Steinhaus*

*property*if no matter how it is placed

*on*

*a*plane, it covers exactly

*one*integer lattice point. Whether or not such

*a*

*set*exists, is an open problem. ... As

*a*corollary, we deduce that closed

*sets*do not

*have*

*the*

*Steinhaus*

*property*, fact noted by Sierpinski [3] under

*the*additional assumption of boundedness. ...

*The*purpose of this paper is to prove

*the*following Theorem. Any

*set*S

*having*

*the*

*Steinhaus*

*property*has empty interior. Our proof requires

*a*number of preliminary lemmas. ...

##
###
A short proof of the converse to a theorem of Steinhaus
[article]

2015
*
arXiv
*
pre-print

*A*result of H.

*Steinhaus*states that any positive Lebesgue measurable

*set*has

*a*

*property*that its difference

*set*contains an open interval around

*the*origin. Y. V. ... Mospan proved that this result is

*the*characterization of absolutely continuous measure. In this note we give

*a*short proof of it. ... Definition 1 . 1

*A*Borel measure µ

*on*R is said to

*have*

*Steinhaus*

*property*(abbreviated SP) if for every µ−measurable

*set*

*A*with µ(

*A*) > 0

*the*difference

*A*−

*A*contains an open interval around zero. ...

##
###
Nonmeasurability in Banach spaces
[article]

2010
*
arXiv
*
pre-print

We show that for

arXiv:1001.0073v1
fatcat:onxq5eaczje67c3c7bo3ihqp2q
*a*σ -ideal with*a**Steinhaus**property*defined*on*Banach space, if two non-homeomorphic Banach with*the*same cardinality of*the*Hamel basis then there is*a*nonmeasurable subset as image ...*A*subset of Z not in I will be called*a*I-positive*set*;*sets*in I will also be called I-null. Also,*the*σ-algebra generated by B(Z) ∪ I will be denoted by B(Z), called*the*I-completion of B(Z). ... Acknowledgement Author is very indebted to Professor Jacek Cichoń for help and critical*remarks*. ...##
###
Measurable Steinhaus sets do not exist for finite sets or the integers in the plane

2017
*
Bulletin of the London Mathematical Society
*

*A*

*Steinhaus*

*set*S ⊆^d for

*a*

*set*

*A*⊆^d is

*a*

*set*such that S has exactly

*one*point in common with τ

*A*, for every rigid motion τ of ^d. ... An old result of Komjáth says that there exists

*a*

*Steinhaus*

*set*for

*A*= ×0 in ^2. We also show here that such

*a*

*set*cannot be Lebesgue measurable. ... Sierpinski [21] showed that

*a*bounded

*set*

*A*which is either closed or open cannot

*have*

*the*lattice

*Steinhaus*

*property*(that is, intersect all rigid motions of Z 2 at exactly

*one*point). ...

##
###
On Steinhaus sets

2005
*
Expositiones mathematicae
*

We give

doi:10.1016/j.exmath.2005.01.011
fatcat:3p4r3ge7tjhtxfslti47njgcxm
*a*common proof of several results*on**Steinhaus**sets*in R d for d 2 including*the*fact that*a**Steinhaus**set*in R 2 must be disconnected. ...*Steinhaus**sets**have*been*the*subject of several recent papers. In this note, we give*a*common proof of some results (which are known in R 2 ) and some new results*on**Steinhaus**sets*in any dimension. ... Let S ⊂ R d be*a**Steinhaus**set*, x ∈ R d and > 0. Then S ∩*A*n (x, ) = ∅ for some n 1.*Remark*. ...##
###
On the Steinhaus property in topological groups

2006
*
Topology and its Applications
*

Let G be

doi:10.1016/j.topol.2005.07.010
fatcat:o6groxauivdn5cg7oc623cc5rm
*a*locally compact Abelian group and μ*a*Haar measure*on*G. ... We prove: (*a*) If G is connected, then*the*complement of*a*union of finitely many translates of subgroups of G with infinite index is μ-thick and everywhere of second category. ... Then*the*σ -ideal N of locally null*sets*with respect to*a*Haar measure*on*G has*the**Steinhaus**property*. ...##
###
The Steinhaus tiling problem and the range of certain quadratic forms
[article]

2000
*
arXiv
*
pre-print

In dimension d=2 (

arXiv:math/0009207v1
fatcat:tylwb4xn3jfbljh7kyferolk2a
*the*original*Steinhaus*problem)*the*question remains open. ... We give*a*short proof of*the*fact that there are no measurable subsets of Euclidean space (in dimension d > 2), which, no matter how translated and rotated, always contain exactly*one*integer lattice point ... Assume from now*on*that*the**set*E is*a**Steinhaus**set*in dimension d. Suppose now that we can find*a*lattice Λ * ⊂ B with det Λ * not an integer. ...##
###
The Steinhaus tiling problem and the range of certain quadratic forms

2002
*
Illinois Journal of Mathematics
*

In dimension d = 2 (

doi:10.1215/ijm/1258130994
fatcat:ghg7oosqrbahjoqpx3incrx46m
*the*original*Steinhaus*problem)*the*question remains open. ... We give*a*short proof of*the*fact that there are no measurable subsets of Euclidean space (in dimension d ≥ 3) which, no matter how translated and rotated, always contain exactly*one*integer lattice point ... Assume from now*on*that*the**set*E is*a**Steinhaus**set*in dimension d. Suppose now that we can find*a*lattice Λ * ⊂ B with det Λ * not an integer. ...##
###
Page 6473 of Mathematical Reviews Vol. , Issue 2001I
[page]

2001
*
Mathematical Reviews
*

*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” forms

*a*tiling of

*the*plane. ... of

*a*plane

*set*with

*a*certain

*property*. ...

##
###
ON THE STEINHAUS TILING PROBLEM FOR Z3

2013
*
Quarterly Journal of Mathematics
*

*Steinhaus*asked in

*the*1950's whether there exists

*a*

*set*in R 2 meeting every isometric copy of Z 2 in precisely

*one*point. Such

*a*"

*Steinhaus*

*set*" was constructed by Jackson and Mauldin. ... Is there

*a*subset S of R 3 meeting every isometric copy of Z 3 in exactly

*one*point? We offer heuristic evidence that

*the*answer is "no". ... However, it is known that no

*Steinhaus*

*set*in R 2 is

*a*Borel

*set*or even has

*the*Baire

*property*[7] . ...

##
###
Beyond Lebesgue and Baire IV: Density topologies and a converse Steinhaus–Weil theorem

2018
*
Topology and its Applications
*

*The*second was our reliance [BinO8]

*on*

*a*localized version of

*the*

*Steinhaus*-Weil

*property*: in S

*the*relative open neighbourhoods of all points were to

*have*

*the*

*Steinhaus*-Weil

*property*. ... (Boundedness of

*a*subadditive function

*on*

*A*and B yields its boundedness

*on*AB and hence

*on*an open

*set*, provided AB has

*the*interior-point

*property*-see §6.9.) ... We thank

*the*Referee for

*a*careful and scholarly reading of

*the*paper, and for some very useful presentational suggestions. ...

##
###
On the Steinhaus tiling problem in three dimensions
[article]

2013
*
arXiv
*
pre-print

*Steinhaus*asked in

*the*1950's whether there exists

*a*

*set*in

*the*plane R^2 meeting every isometric copy of Z^2 in precisely

*one*point. Such

*a*"

*Steinhaus*

*set*" was constructed by Jackson and Mauldin. ... Is there

*a*subset of R^3 meeting every isometric copy of Z^3 in exactly

*one*point? We offer heuristic evidence that

*the*answer is "no". ... However, it is known that no

*Steinhaus*

*set*in R 2 is

*a*Borel

*set*or even has

*the*Baire

*property*[7] . ...

##
###
Beyond Lebesgue and Baire IV: Density topologies and a converse Steinhaus-Weil Theorem
[article]

2017
*
arXiv
*
pre-print

*The*

*Steinhaus*-Weil interior-points theorem ('

*on*AA^-1') plays

*a*crucial role here; so too does its converse,

*the*Simmons-Mospan theorem. ...

*One*can often handle

*the*(Baire) category case and

*the*(Lebesgue, or Haar) measure cases together, by working bi-topologically: switching between

*the*original topology and

*a*suitable refinement (

*a*density ... We thank

*the*Referee for

*a*careful and scholarly reading of

*the*paper, and for some very useful presentational suggestions. ...

##
###
Page 8683 of Mathematical Reviews Vol. , Issue 2004k
[page]

2004
*
Mathematical Reviews
*

*A*

*set*with this

*property*is called

*a*

*Steinhaus*

*set*.

*The*relation of

*Steinhaus*

*sets*to tiling is clear.

*A*

*set*S is

*a*

*Steinhaus*

*set*if and only if every rotation of S tiles

*the*plane. ... Recently

*the*authors of

*the*present paper

*have*answered this long-standing question of

*Steinhaus*by showing that there is

*a*

*Steinhaus*

*set*. This solution was published in [J. Amer. Math. ...

##
###
On the difference property of families of measurable functions

2003
*
Colloquium Mathematicum
*

We show that, generally, families of measurable functions do not

doi:10.4064/cm97-2-4
fatcat:fjuufahwrzhpnnmtahcp5zih74
*have**the*difference*property*under some assumption. ... We also show that there are natural classes of functions which do not*have**the*difference*property*in ZFC. This extends*the*result of Erdős concerning*the*family of Lebesgue measurable functions. ... First, we*have**the*well known theorems which explain*the*names of*the*Ostrowski*property*and*Steinhaus**property*. They concern Lebesgue measure*on*R. ...
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