I think the Besicovitch sets are really unexpected, and then also Kakeya sets. The first type of sets are sets of measure 0 in the plane, with a line segment of unit length in every direction. The latter sets are sets where a needle of unit length can be rotated a full turn (moving back and forth is also admitted). Kakeya sets are naturally Besicovitch sets, and surprisingly, there isa Kakeya set of any positive Lebegue measure.
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