## On the multiparameter Falconer distance problem

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- by Xiumin Du, Yumeng Ou and Ruixiang Zhang PDF
- Trans. Amer. Math. Soc.
**375**(2022), 4979-5010 Request permission

## Abstract:

We study an extension of the Falconer distance problem in the multiparameter setting. Given $\ell \geq 1$ and $\mathbb {R}^{d}=\mathbb {R}^{d_1}\times \cdots \times \mathbb {R}^{d_\ell }$, $d_i\geq 2$. For any compact set $E\subset \mathbb {R}^{d}$ with Hausdorff dimension larger than $d-\frac {\operatorname {min}(d_i)}{2}+\frac {1}{4}$ if $\operatorname {min}(d_i)$ is even, $d-\frac {\operatorname {min}(d_i)}{2}+\frac {1}{4} +\frac {1}{4\operatorname {min}(d_i)}$ if $\operatorname {min}(d_i)$ is odd, we prove that the multiparameter distance set of $E$ has positive $\ell$-dimensional Lebesgue measure. A key ingredient in the proof is a new multiparameter radial projection theorem for fractal measures.## References

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## Additional Information

**Xiumin Du**- Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
- MR Author ID: 1234739
- ORCID: setImmediate$0.9884010903214075$2
**Yumeng Ou**- Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
- MR Author ID: 1112799
**Ruixiang Zhang**- Affiliation: Department of Mathematics, University of California, Berkeley, Berkeley, California 94720
- MR Author ID: 942634
- Received by editor(s): June 24, 2021
- Received by editor(s) in revised form: December 28, 2021
- Published electronically: May 4, 2022
- Additional Notes: The first author was supported by NSF DMS-2107729. The second author was supported by NSF DMS-2042109. The third author was supported by the NSF grant DMS-1856541, DMS-1926686 and by the Ky Fan and Yu-Fen Fan Endowment Fund at the Institute for Advanced Study.
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 4979-5010 - MSC (2020): Primary 42B20, 28A80
- DOI: https://doi.org/10.1090/tran/8667
- MathSciNet review: 4439497