Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 26J / f

Solution

Let

∣ t ∣ < 2 α

. Both

E

and its translate

E + t

lie in an interval of length

1 + ∣ t ∣

m (E \cap (E + t)) = m (E) + m (E + t) - m (E \cup (E + t)) \geq 2 (\frac{1}{2} + α) - (1 + ∣ t ∣) = 2 α - ∣ t ∣ > 0.

Choose

x \in E \cap (E + t)

. Then

x \in E

and

x - t \in E

, so

t = x - (x - t) \in E - E

(- 2 α, 2 α) \subseteq E - E .

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