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analysis

Source: miklos schweitzer 1994 q6

October 16, 2021
real analysisMeasure theory

Problem Statement

Show that if n is an arbitrary natural number and n≤K≤n2\sqrt n \leq K \leq \frac{n}{2}, then there exist n distinct integers, kjk_j ( j = 1, ..., n ) such that ∣∑j=1neikjt∣≥K\bigg | \sum_ {j = 1} ^ ne ^ {ik_jt} \bigg | \geq K is satisfied on a subset of the interval (−π,π)(- \pi, \pi) with Lebesgue measure at least cnK2\frac{cn}{K^2} , where c is a suitable absolute constant.
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