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We prove that, for every 0 ⩽ s ⩽ 1, there is a Hamel basis of the vector space of reals over the field of rationals that has Hausdorff dimension s. The logic of our proof is of particular interest. The statement of our theorem is classical; it does not involve the theory of computing. However, our proof makes essential use of algorithmic fractal dimension–a computability-theoretic construct–and the point-to-set principle of J. Lutz and N. Lutz (2018). 

Abstract Supported nanoparticles are broadly employed in industrial catalytic processes, where the active sites can be tuned by metal-support interactions (MSIs). Although it is well accepted that supports can modify the chemistry of metal nanoparticles, systematic utilization of MSIs for achieving desired catalytic performance is still challenging. The developments of supports with appropriate chemical properties and identification of the resulting active sites are the main barriers. Here, we develop two-dimensional transition metal carbides (MXenes) supported platinum as efficient catalysts for light alkane dehydrogenations. Ordered Pt 3 Ti and surface Pt 3 Nb intermetallic compound nanoparticles are formed via reactive metal-support interactions on Pt/Ti 3 C 2 T x and Pt/Nb 2 CT x catalysts, respectively. MXene supports modulate the nature of the active sites, making them highly selective toward C–H activation. Such exploitation of the MSIs makes MXenes promising platforms with versatile chemical reactivity and tunability for facile design of supported intermetallic nanoparticles over a wide range of compositions and structures.