On the projection of irregular linearly measurable plane sets of points

Extract

1. Elsewhere I have expressed the opinion that a linearly measurable plane set of points, of upper density ½, has zero projection (i.e. projection of measure zero) on almost all directions. Though unable to prove this I have there managed to obtain a partial result. I would not attempt to guess to what extent this is true of a general irregular linearly measurable plane set; these sets are not amenable to the methods used in the case of those of upper density ½, the latter having a remarkable and simple structure. It is a quite trivial matter to construct an irregular set, of upper density 1, with positive projection on some one direction; and from it we can deduce another such set with positive projections on a denumerable every-where-dense set of directions. But, working on a suggestion due to Mr Besicovitch, I have obtained an irregular linearly measurable plane set, of upper density 1, with positive projection on a set P of directions, where the members of P in any sector have cardinal t, though P has measure zero. It may be that the set of points has positive projection on directions other than those of P, but there are no obvious indications of any such directions.