Local dimensions of sliced measures and stability of packing dimensions of sections of sets

Let m and n be integers with 0<m<n. We relate the absolutely continuous and singular parts of a measure μ on to certain properties of plane sections of μ. This leads us to prove, among other things, that the lower local dimension of (n−m)-plane sections of μ is typically constant provided that the Hausdorff dimension of μ is greater than m. The analogous result holds for the upper local dimension if μ has finite t-energy for some t>m. We also give a sufficient condition for stability of packing dimensions of section of sets.