M.Sc Thesis


M.Sc StudentCarmel Amir
SubjectOn the Odd Area of Discs and Arcs
DepartmentDepartment of Mathematics
Supervisor PROF. Rom Pinchasi


Abstract

The odd area of a collection of measurable sets is a highly compelling geometric problem that attracts broad research in recent years. The problem is formulated as follows.
For a collection F of measurable sets in the Euclidean plane, the odd area of F is defined to be the Lebesgue measure of all the points that belong to an odd number of members of F. Now, fix a measurable set X and consider any collection
F of odd cardinality of translated copies of X.
What is the minimum odd area covered by F, over all such collections
F?


Although immense study this problem remains open for circular discs.

In this thesis we show novel results regarding the odd area of circular discs. Moreover, we suggest another formulation for the odd area of shapes on a torus, rather than on the Euclidean plane. In this framework we managed to fully nail down the odd area of boxes in an n-dimensional torus. We also present a new technique to evaluate the odd area of a collection of measurable sets which can be further utilized to provide a new original proof for the classic problem regarding the odd area of unit segments on the real line.