A REMARK ON QUASICONFORMAL DIMENSION DISTORTION ON THE LINE

Abstract. The general dimension distortion result of Astala says that a one dimensional set goes to a set of dimension at least 1 − k under a k-quasiconformal mapping. An improved version for rectifiable sets appears in recent work of Astala, Clop, Mateu, Orobitg and Uriarte-Tuero in connection with quasiregular removability problems. We give an alternative proof of their result establishing a bound of the form 1 − ck2, provided that either the initial or the target set lies on a straight line. The bound 1 − k2 holds under the additional assumption that the line stays fixed. 1