Radial projection theorems in finite spaces

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Motivated by recent results on radial projections and applications to the celebrated Falconer distance problem, we study radial projections in the setting of finite fields. More precisely, we extend results due to Mattila and Orponen [Proc. Amer. Math. Soc. 144 (2016), pp. 3419–3430], Orponen [Bull. Lond. Math. Soc. 50 (2018), pp. 1–6], and Liu [Rev. Mat. Iberoam. 37 (2020), pp. 1307–1319] to finite spaces. In some cases, our results are stronger than the corresponding results in the continuous setting. In particular, we solve the finite field analog of a conjecture due to Liu and Orponen on the exceptional set of radial projections of a set of dimension between d − 2 d-2 and d − 1 d-1 .