Geodesic
The problem on the measure of the union of line segments in the plane with restrictions on the set of their ends
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 2, pp. 1319-1331

Voir la notice de l'article provenant de la source Math-Net.Ru

Some time ago M.A. Patrakeev asked the following question. Let $A$ and $B$ be zero-measure subsets of the unit segment. Let $\varphi$ be bijection between $A$ and $B$. Denote by $S(A,B,\varphi)$ the union of all segments in the plane with the endpoints $(a,0)$ and $(\varphi(a),1)$ for some $a\in A$. The question is what the measure of the set $S(A,B,\varphi)$. We answer this question.
Keywords: measure, plane. 
@article{SEMR_2021_18_2_a71,
     author = {A. E. Lipin},
     title = {The problem on the measure of the union of line segments in the plane with restrictions on the set of their ends},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1319--1331},
     publisher = {mathdoc},
     volume = {18},
     number = {2},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2021_18_2_a71/}
}
TY  - JOUR
AU  - A. E. Lipin
TI  - The problem on the measure of the union of line segments in the plane with restrictions on the set of their ends
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2021
SP  - 1319
EP  - 1331
VL  - 18
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2021_18_2_a71/
LA  - ru
ID  - SEMR_2021_18_2_a71
ER  - 
%0 Journal Article
%A A. E. Lipin
%T The problem on the measure of the union of line segments in the plane with restrictions on the set of their ends
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2021
%P 1319-1331
%V 18
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2021_18_2_a71/
%G ru
%F SEMR_2021_18_2_a71
A. E. Lipin. The problem on the measure of the union of line segments in the plane with restrictions on the set of their ends. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 2, pp. 1319-1331. http://geodesic.mathdoc.fr/item/SEMR_2021_18_2_a71/