Geodesic
A Set of Plane Measure Zero Containing all Finite Polygonal Arcs
Canadian journal of mathematics, Tome 22 (1970) no. 4, pp. 815-821

Voir la notice de l'article provenant de la source Cambridge University Press

We say a (plane) set A contains all sets of some type if, for each B of type , there is a subset of A that is congruent to B. Recently, Besicovitch and Rado [3] and independently, Kinney [5] have constructed sets of plane measure zero containing all circles. In these papers it is pointed out that the set of all similar rectangles, some sets of confocal conies and other such classes of sets can be contained in sets of plane measure zero, but all these generalizations rely in some way on the symmetry, or similarity of the sets within the given type.In this paper we construct a set of plane measure zero containing all finite polygonal arcs (i.e., the one-dimensional boundaries of all polygons with a finite number of sides) with slightly stronger results if we restrict our attention to k-gons for some fixed k. 
Ward, D. J. A Set of Plane Measure Zero Containing all Finite Polygonal Arcs. Canadian journal of mathematics, Tome 22 (1970) no. 4, pp. 815-821. doi: 10.4153/CJM-1970-091-4
@article{10_4153_CJM_1970_091_4,
     author = {Ward, D. J.},
     title = {A {Set} of {Plane} {Measure} {Zero} {Containing} all {Finite} {Polygonal} {Arcs}},
     journal = {Canadian journal of mathematics},
     pages = {815--821},
     year = {1970},
     volume = {22},
     number = {4},
     doi = {10.4153/CJM-1970-091-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1970-091-4/}
}
TY  - JOUR
AU  - Ward, D. J.
TI  - A Set of Plane Measure Zero Containing all Finite Polygonal Arcs
JO  - Canadian journal of mathematics
PY  - 1970
SP  - 815
EP  - 821
VL  - 22
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1970-091-4/
DO  - 10.4153/CJM-1970-091-4
ID  - 10_4153_CJM_1970_091_4
ER  - 
%0 Journal Article
%A Ward, D. J.
%T A Set of Plane Measure Zero Containing all Finite Polygonal Arcs
%J Canadian journal of mathematics
%D 1970
%P 815-821
%V 22
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1970-091-4/
%R 10.4153/CJM-1970-091-4
%F 10_4153_CJM_1970_091_4

[1] 1. Besicovitch, A. S., On linear sets of points of fractional dimension, Math. Ann. 101 (1929), 161–193. Google Scholar

[2] 2. Besicovitch, A. S. and Moran, P. A. P., The measure of product and cylinder sets, J. London Math. Soc. 20 (1945), 110–120. Google Scholar

[3] 3. Besicovitch, A. S. and Rado, R., A plane set of measure zero containing circumferences of every radius, J. London Math. Soc. 43 (1968), 717–719. Google Scholar

[4] 4. Davies, R. O., Marstrand, J. M., and Taylor, S. J., On the intersections of transforms of linear sets, Colloq. Math. 7 (1959/60), 237–243. Google Scholar

[5] 5. Kinney, J. R., A thin set of circles, Amer. Math. Monthly 75 (1968), 1077–1081. Google Scholar

Cité par Sources :