Abstract
We develop the methods used by Rudnev and Wheeler (2022) to prove an incidence theorem between arbitrary sets of Möbius transformations and point sets in \({\mathbb {F}}_p^2\). We also note some asymmetric incidence results, and give applications of these results to various problems in additive combinatorics and discrete geometry. For instance, we give an improvement to a result of Shkredov concerning the number of representations of a non-zero product defined by a set with small sum-set, and a version of Beck’s theorem for Möbius transformations.
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Notes
We use the standard Vinogradov notation \(X \ll Y\) to mean that there exists an absolute constant C with \(X \le CY\). We have \(Y \gg X\) iff \(X \ll Y\). We write \(X\sim Y\) to mean that \(X \ll Y\) and \(Y \ll X\).
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Acknowledgements
The first author was supported by Austrian Science Fund FWF grant P-34180. The second author was funded by the EPSRC and University of Bristol. We thank Oliver Roche-Newton, Misha Rudnev, and Sophie Stevens for helpful suggestions and conversations. We also thank the anonymous reviewer for their helpful comments.
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Warren, A., Wheeler, J. Incidences of Möbius Transformations in \({\mathbb {F}}_p\). Discrete Comput Geom 70, 1025–1037 (2023). https://doi.org/10.1007/s00454-022-00442-4
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DOI: https://doi.org/10.1007/s00454-022-00442-4