The Hausdorff dimension of graphs of Weierstrass functions

Hunt, Brian

doi:10.1090/S0002-9939-98-04387-1

The Hausdorff dimension of graphs of Weierstrass functions
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Proc. Amer. Math. Soc. 126 (1998), 791-800 Request permission

Abstract:

The Weierstrass nowhere differentiable function, and functions constructed from similar infinite series, have been studied often as examples of functions whose graph is a fractal. Though there is a simple formula for the Hausdorff dimension of the graph which is widely accepted, it has not been rigorously proved to hold. We prove that if arbitrary phases are included in each term of the summation for the Weierstrass function, the Hausdorff dimension of the graph of the function has the conjectured value for almost every sequence of phases. The argument extends to a much wider class of Weierstrass-like functions.

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