Arithmetic representations of real numbers in terms of self-similar sets
Kan Jiang, Lifeng Xi · Annales Academiae Scientiarum Fennicae Mathematica · 2019
Suppose n ≥ 2 andIn this paper, we analyze the Hausdorff dimension and Hausdorff measure of the following setwhere Card(S x ) denotes the cardinality of S x , and r ∈ N + .We prove under the so-called covering condition that the Hausdorff dimension of U 1 can be calculated in terms of some matrix.Moreover, if r ≥ 2, we also give some sufficient conditions such that the Hausdorff dimension of U r takes only finite values, and these values can be calculated explicitly.Furthermore, we come up with some sufficient conditions such that the dimensional Hausdorff measure of U r is infinity.Various examples are provided.Our results can be viewed as the exceptional results for the classical slicing problem in geometric measure theory.