Abstract
We introduce two notions of fractal sumset properties. A compact set \(K\subset\mathbb{R}^d\) is said to have the Hausdorff sumset property (HSP) if for any \(\ell\in\mathbb{N}_{\ge 2}\) there exist compact sets \(K_1,K_2\),..., \(K_\ell\) such that \(K_1+K_2+\cdots+K_\ell\subset K\) and \(\dim_H K_i=\dim_H K\) for all \(1\le i\le \ell\). Analogously, if we replace the Hausdorff dimension by the packing dimension in the definition of HSP, then the compact set \(K\subset\mathbb{R}^d\) is said to have the packing sumset property (PSP). We show that the HSP fails for certain homogeneous self-similar sets satisfying the strong separation condition, while the PSP holds for all homogeneous self-similar sets in \(\mathbb{R}^d\).
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Acknowledgements
The authors thank the anonymous referees for many useful suggestions which improve the presentation of the paper. The first author wants to thank Tuomas Orpenon for many useful discussions during the Fractal Geometry conference in Edinburgh 2023, especially for Remark 1.2 and many useful references.
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The second author was supported by NSFC No. 12071148, and Science and Technology Commission of Shanghai Municipality (STCSM) No. 22DZ2229014, and Fundamental Research Funds for the Central Universities No. YBNLTS2023-016.
The first author was supported by NSFC No. 11971079.
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Kong, D., Wang, Z. Fractal sumset properties. Acta Math. Hungar. (2024). https://doi.org/10.1007/s10474-024-01421-2
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DOI: https://doi.org/10.1007/s10474-024-01421-2