Abstract
We prove that the number of directions contained in a set of the form A × B ⊂ AG(2,p), where p is prime, is at least |A||B| − min{|A|, |B|} + 2. Here A and B are subsets of GF(p) each with at least two elements and |A||B| <p. This bound is tight for an infinite class of examples. Our main tool is the use of the Rédei polynomial with Szőnyi’s extension. As an application of our main result, we obtain an upper bound on the clique number of a Paley graph, matching the current best bound obtained recently by Hanson and Petridis.
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Acknowledgements
The research of the first author was supported in part by a Four Year Doctoral Fellowship from the University of British Columbia. The research of the second author was supported in part by an NSERC Discovery grant and OTKA K 119528 grant. The work of the second author was also supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 741420, 617747, 648017). The research of the third author was supported in part by Killam and NSERC doctoral scholarships. We also thank Sammy Luo for helpful comments.
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Di Benedetto, D., Solymosi, J. & White, E.P. On the Directions Determined by a Cartesian Product in an Affine Galois Plane. Combinatorica 41, 755–763 (2021). https://doi.org/10.1007/s00493-020-4516-z
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DOI: https://doi.org/10.1007/s00493-020-4516-z