MathDB

Set covers <s_n fraction of interval

Source: Miklós Schweitzer 2017, problem 7

January 13, 2018

measureMeasure theoryintervalreal analysiscollege contestsMiklos Schweitzer

Problem Statement

Characterize all increasing sequences

(s_n)

of positive reals for which there exists a set

A\subset \mathbb{R}

with positive measure such that

\lambda(A\cap I)<\frac{s_n}{n}

holds for every interval

I

with length

1/n

, where

\lambda

denotes the Lebesgue measure.