Generalized Minkowski Content and the Vibrations of Fractal Drums and Strings

In [La1], the second author has obtained a sharp error estimate for the eigenvalue distribution of the Laplacian on bounded open sets $\Omega \subset {\rm {\bf R}}^n$ with fractal boundaries (i.e., `fractal drums'). Further, he and Pomerance [LaPo1,2] studied in detail the case of `fractal strings' (i.e., $n=1$) and established in the process some unexpected connections with the Riemann zeta-function $\zeta = \zeta(s)$ in the `critical strip' $0 <Re \ s <1$. Later on, still when $n=1$, Lapidus and Maier [LaMa1,2] obtained a new characterization of the Riemann hypothesis by means of an associated inverse spectral problem. In this paper, we will extend most of these results by using, in particular, the notion of generalized Minkowski content which is defined through some suitable `gauge functions' other than the power functions. In the situation when the power function is not the natural `gauge function', this will enable us to obtain more precise estimates, with a broader potential range of applications than in the above papers. Complete proofs of the results announced here will be provided in [HeLa].

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Published 1 January 1996