By K. J. FalconerThis ebook features a rigorous mathematical remedy of the geometrical facets of units of either critical and fractional Hausdorff measurement. Questions of neighborhood density and the lifestyles of tangents of such units are studied, in addition to the dimensional houses in their projections in quite a few instructions. in terms of units of essential measurement the dramatic changes among typical 'curve-like' units and abnormal 'dust like' units are exhibited. the idea is said by way of duality to Kayeka units (sets of 0 quarter containing traces in each direction). the ultimate bankruptcy contains various examples of units to which the final idea is acceptable: discussions of curves of fractional measurement, self-similar units, unusual attractors, and examples from quantity conception, convexity etc. there's an emphasis at the simple instruments of the topic akin to the Vitali masking lemma, web measures and Fourier remodel equipment.