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Consider a standard Cantor set in the plane of Hausdorff dimension 1. If the linear density of the associated measure $\mu$ vanishes, then the set of points where the principal value of the Cauchy singular integral of $\mu$ exists has Hausdorff dimension 1. The result is extended to Cantor sets in $\mathbb{R}^d$ of Hausdorff dimension $\alpha$ and Riesz singular integrals of homogeneity $-\alpha$, 0 < $\alpha$ < d : the set of points where the principal value of the Riesz singular integral of $\mu$ exists has Hausdorff dimension $\alpha$. A martingale associated with the singular integral is introduced to support the proof., Comment: 13 pages

The project is a combination of practice as research based investigation, and material outcomes in the form of steel objects. The process of this research, which included travelling to the United States to interact with my contemporaries and the steel craft industry, highlighted for me the interconnectivity of three elements of making. These three elements articulate my experiences of practice as research, in a textual form, and add to the explanatory component of my project; they are: Lineage: The passing on of knowledge and skills, the lineage from one maker to another. This is physically described in Dogon (Mali) mythology as passing a piece of iron from the blacksmith to the student's hammer through the generations. It is alternatively articulated through the increasing transmission of skills via online forums, video demonstration and social media. Material/medium and process: Form development specific to the material, composition, problem solving and intuitive interaction; the conception of tools to manipulate and transform steel. This is about what can actually be done, but also about perception of the material because using steel means always working with the paradox of rigid and near fluid states. The social implications of being a maker in the 21st century: In situating my work I am not set within a cottage industry, although the working of steel stems from a preindustrial craft that developed into a large industry. The skills and techniques that I use are from both eras. Craft ideology and making are now a post-industrial endeavour. My practice is a hybrid: it embraces industrial skilling, yet is translated through a crafts and do-it-yourself approach. The new work produced during this project is the research outcome, a physical interpretation of the knowledge and experience gained. It is also a continuation of the research through making, a thread in the larger fabric of practice. The objects made are complex compositions utilising a number of forged and fabricated forms. The objects have a level of ambiguous utility, inviting the viewer with an aspect of familiarity, and the potential for possible human interaction, freeing the objects of specific classification.