Abstract
In this paper, we study dot-product sets and k-simplices in \(\mathbb {Z}_n^d\) for odd n, where \(\mathbb {Z}_n\) is the ring of residues modulo n. We show that if E is sufficiently large then the dot-product set of E covers the whole ring. In higher dimensional cases, if E is sufficiently large then the set of simplices and the set of dot-product simplices determined by E, up to congurence, have positive densities.
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Acknowledgements
This research is funded by the VNU University of Science, Vietnam National University, Hanoi under project number TN.21.05. Nguyen Van The was funded by Vingroup JSC and supported by the Master, PhD Scholarship Programme of Vingroup Innovation Foundation (VINIF), Institute of Big Data, code VINIF.2021.ThS.KHTN.01.
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Communicated by Alex Iosevich.
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The, N.V., Vinh, L.A. Dot-Product Sets and Simplices Over Finite Rings. J Fourier Anal Appl 28, 38 (2022). https://doi.org/10.1007/s00041-022-09933-7
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DOI: https://doi.org/10.1007/s00041-022-09933-7