MathDB

A tile

T

is a union of finitely many pairwise disjoint arcs of a unit circle

K

. The size of

T

, denoted by

|T|

, is the sum of the lengths of the arcs

T

consists of, divided by

2\pi

. A copy of

T

is a tile

T'

obtained by rotating

T

about the centre of

K

through some angle. Given a positive real number

\varepsilon < 1

, does there exist an infinite sequence of tiles

T_1,T_2,\ldots,T_n,\ldots

satisfying the following two conditions simultaneously: 1)

|T_n| > 1 - \varepsilon

for all

n

; 2) The union of all

T_n'

(as

n

runs through the positive integers) is a proper subset of

K

for any choice of the copies

T_1'

T_2'

\ldots

T_n', \ldots

?
In the extralist the problem statement had the clause "three conditions" rather than two, but only two are presented, the ones you see. I am quite confident this is a typo or that the problem might have been reformulated after submission.

Problem Statement