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We show that a compact, connected set which has uniform oscillations at all points and at all scales has dimension strictly larger than 1. We also show that limit sets of certain Kleinian groups have this property. More generally, we show that if$G$is a non-elementary, analytically finite Kleinian group, and its limit set Λ($G$) is connected, then Λ($G$) is either a circle or has dimension strictly bigger than 1.

In this paper, we study the decomposition of sets of consecutive integers, i.e., how to express the set {0, 1, . . . ,mn−1} as the Minkovski sum of two sets, one with m terms and the other with n terms. First, we translate the problem into the language of polynomials. Then we use the properties of cyclotomic polynomials to determine the structure of all the solutions. Then, we deduce formulas for the number of such expressions. Finally, we give an algorithm to calculate the number of such expressions and analyze its computational complexity.

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For any unital separable simple infinite-dimensional nuclear$C$^{∗}-algebra with finitely many extremal traces, we prove that $$ \mathcal{Z} $$ -absorption, strict comparison and property (SI) are equivalent. We also show that any unital separable simple nuclear$C$^{∗}-algebra with tracial rank zero is approximately divisible, and hence is $$ \mathcal{Z} $$ -absorbing.