arXiv:1203.5296v1 [math.CA] 23 Mar 2012 HAUSDORFF DIMENSION AND NON-DEGENERATE FAMILIES OF PROJECTIONS ESA J ¨ ARVENP ¨ A ¨ A 1 , MAARIT J ¨ ARVENP ¨ A ¨ A 2 , AND TAM ´ AS KELETI 3 Abstract. We study parametrized families of orthogonal projections for which the dimension of the parameter space is strictly less than that of the Grass- mann manifold. We answer the natural question of how much the Hausdorff dimension may decrease by verifying the best possible lower bound for the dimension of almost all projections of a finite measure. We also show that a similar result is valid for smooth families of maps from n-dimensional Eu- clidean space to m-dimensional one. 1. Introduction The behaviour of different concepts of dimensions of sets and measures under projections has been investigated intensively for several decades. The study was initiated by Marstrand [Mar] in the 1950’s. Mattila [Mat1] considered Haus- dorff dimension of sets in the 1970’s, and in the late 1980’s and in the 1990’s several authors contributed to the field. In 2000 Peres and Schlag [PS] proved a very general result concerning transversal families of mappings and Sobolev dimension. For a more detailed account of the history, see the survey of Mattila [Mat3]. All the above results concerning Hausdorff dimension may be simplified by stating that the dimension is preserved under almost all projections. The essen- tial assumption is transversality which is guaranteed in many cases by identifying the parameter space with an open subset of the Grassmann manifold. The ques- tion we are addressing is that how much the dimension may drop under almost all projections provided that the dimension of the parameter space is less than that of the Grassmann manifold. The following conclusion can be drawn from [PS]: Fubini’s theorem implies that for a given set or a measure the dimension is preserved for almost all projections in almost all k-dimensional families for any k. Hence, for a given measure one obtains information for typical families. However, in general there is no way to conclude whether a given family is typical for a given measure. Furthermore, the results of [PS] concerning exceptional sets 2000 Mathematics Subject Classification. 28A80, 37C45. Key words and phrases. Projection, Hausdorff dimension, measure. We acknowledge the support of the Centre of Excellence in Analysis and Dynamics Research funded by the Academy of Finland. The third author was also supported by OTKA grant no. 72655 and J´ anos Bolyai Fellowship. 1