Annales Academiæ Scientiarum Fennicæ

Mathematica

Volumen 32, 2007, 545-548

# SETS OF FINITE *H*^{1} MEASURE THAT
INTERSECT POSITIVELY MANY LINES
IN INFINITELY MANY POINTS

## Marianna Csörnyei and David Preiss

University College London, Department of Mathematics

Gower Street, London WC1E 6BT, UK; mari 'at' math.ucl.ac.uk

University of Warwick, Mathematics Institute

Coventry, CV4 7AL, UK; D.Preiss 'at' warwick.ac.uk

**Abstract.**
We construct a planar Borel set
*A* of finite *H*^{1}-measure such that through
positively many points of *A*, positively many lines meet
*A* at infinitely
many points. This answers a question of Mattila.

**2000 Mathematics Subject Classification:**
Primary 28A75.

**Key words:**
Besicovitch's projection theorem,
rectifiability, duality.

**Reference to this article:** M. Csörnyei and D. Preiss:
Sets of finite *H*^{1} measure that intersect
positively many lines in infinitely many points
Ann. Acad. Sci. Fenn. Math. 32 (2007), 545-548.

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