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This paper investigates the spectral properties of Jacobi matrices withlimit-periodic coefficients. We show that for a residual set of such matrices,the spectrum is a Cantor set of zero Lebesgue measure, and the spectralmeasures are purely singular continuous. For a dense set of limit-periodicJacobi matrices we can strengthen the result and show that the spectrum is aCantor set of zero lower box counting dimension, and hence in particular ofzero Hausdorff dimension, while still retaining the singular continuity of thespectral type. We also show how results of this nature can be established byfixing the off-diagonal coefficients and varying only the diagonalcoefficients, and, in a more restricted version, by fixing the diagonalcoefficients to be zero and varying only the off-diagonal coefficients. Weapply these results to produce examples of weighted Laplacians on themultidimensional integer lattice having purely singular continuous spectraltype and zero-dimensional spectrum.

Thin Spectra and Singular Continuous Spectral Measures for Limit-Periodic Jacobi Matrices