AbstractThis paper investigates the spectral properties of Jacobi matrices with limitâperiodic coefficients. We show that generically the spectrum is a Cantor set of zero Lebesgue measure, and the spectral measures are purely singular continuous. For a dense set of limitâperiodic Jacobi matrices, we show that the spectrum is a Cantor set of zero lower box counting dimension while still retaining the singular continuity of the spectral type. We also show how results of this nature can be established by fixing the offâdiagonal coefficients and varying only the diagonal coefficients, and, in a more restricted version, by fixing the diagonal coefficients to be zero and varying only the offâdiagonal coefficients. We apply these results to produce examples of weighted Laplacians on the multidimensional integer lattice having purely singular continuous spectral type and zeroâdimensional spectrum.