Distinct Distances with $\ell_p$ Spaces

Moaaz AlQady; Riley Chabot; William Dudarov; Linus Ge; Mandar Juvekar; Srikanth Kundeti; Neloy Kundu; Kevin Lu; Yago Moreno; Sibo Peng; Samuel Speas; Julia Starzycka; Henry Steinthal; Anastasiia Vitko

doi:10.1016/j.comgeo.2021.101785

arXiv · 2010.05184

Distinct Distances with $\ell_p$ Spaces

Moaaz AlQady·Riley Chabot·William Dudarov·Linus Ge·Mandar Juvekar·Srikanth Kundeti·Neloy Kundu·Kevin Lu·Yago Moreno·Sibo Peng·Samuel Speas·Julia Starzycka·Henry Steinthal·Anastasiia Vitko

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Abstract

We study Erd\H os's distinct distances problem under $\ell_p$ metrics with integer $p$. We improve the current best bound for this problem from $Ω(n^{4/5})$ to $Ω(n^{6/7-ε})$, for any $ε>0$. We also characterize the sets that span an asymptotically minimal number of distinct distances under the $\ell_1$ and $\ell_\infty$ metrics.

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Moaaz AlQady, Riley Chabot, William Dudarov, Linus Ge, Mandar Juvekar, Srikanth Kundeti, Neloy Kundu, Kevin Lu, Yago Moreno, Sibo Peng, Samuel Speas, Julia Starzycka, Henry Steinthal, Anastasiia Vitko. 2020-10-11. Distinct Distances with $\ell_p$ Spaces. https://doi.org/10.1016/j.comgeo.2021.101785

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