Abstract
It is shown that for any \(\alpha \in ]\frac{1}{2},1[\) there exists a symmetric probability measure \(\sigma \) on the torus such that the Hausdorff dimension of its support is \(\alpha \) and \(\sigma *\sigma \) is absolutely continuous with flat continuous Radon-Nikodym derivative. Namely, we obtain a symmetric version of Saeki’s theorem but the flat Radon-Nikodym derivative of \(\sigma *\sigma \) cannot be a Lipschitz function.
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Communicated by Rosihan M. Ali.
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el Abdalaoui, e.H. On the Symmetric Version of Saeki’s Theorem and Flat Densities. Bull. Malays. Math. Sci. Soc. 46, 128 (2023). https://doi.org/10.1007/s40840-023-01525-y
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DOI: https://doi.org/10.1007/s40840-023-01525-y
Keywords
- Symmetric measure
- Singular measure
- Convolution
- Hausdorff dimension
- Distance of Hausdorff
- Saeki’s theorem, flat densities