Number Theory

Volume 1135 of the series Lecture Notes in Mathematics pp 1-8


Hausdorff dimension of sets arising in number theory

  • Richard T. BumbyAffiliated withRutgers University

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The sets considered here have their roots in arithmetic, but the theoretical tools introduced to compute their Hausdorff dimensions should have broader interest and application. In particular, the relation of the Hausdorff dimension to the spectral radius of the subdivision operator provides a means of eliminating ad hoc estimates, thereby sharpening the calculations. The use of monotonicity to allow inequalities of functions to be tested by finite numerical calculations does not seem to have a place in the numerical analysis arsenal. It bears further study. The "spectral analysis" given by equation (4) illustrate a self-duality which seems to be present also for the circle-packing example. This is likely to be an important structure.