Literatur
S. Saks, Théorie de l'intégrale, p. 169–175.
Long ago I thought that this theorem was true and I suggested to Mr Gillis that he should investigate from this side sets of upper density 1/2. By an ingenious method Gillis arrived at some partial results concerning this class of sets. He has also studied sets of directions from which an irregular set can be projected into a set of positive measure and he has constructed a set for which the set of such directions has the power of continuum in any angle, however small. I have quoted his papers in my paper II.
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Paper (II) under the same title has been published in Math. Annalen115 (1938). The literature to the question is given there.
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Besicovitch, A.S. On the fundamental geometrical properties of linearly measurable plane sets of points (III). Math. Ann. 116, 349–357 (1939). https://doi.org/10.1007/BF01597361
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DOI: https://doi.org/10.1007/BF01597361