On an analogue of BRK-type sets in finite fields

A Besicovitch-Rado-Kinney (BRK) set in R 𝑛 contains a hypersphere of every radius.In F 𝑛 𝑞 , BRK-type sets of degree ℓ analogously contain a family of (𝑛-1)-dimensional surfaces, parametrized by a dilation factor and determined by a fixed homogeneous polynomial of degree ℓ.We define (𝑛, 𝑑)-BRK-type sets of degree ℓ, which contain a family of 𝑑-dimensional sets parametrized by an (𝑛 -𝑑)-dimensional dilation factor and determined by fixed homogeneous polynomials of degree ℓ.We use the polynomial method to obtain a lower bound |𝑆 | ≳ 𝑛,ℓ 𝑞 𝑛 on (𝑛, 𝑑)-BRK-type sets 𝑆 of degree ℓ.We obtain an improved lower bound |𝑆 | ≥ (𝑞-1) 𝑛 (ℓ+1-2ℓ/𝑞) 𝑛 by implementing the method of multiplicities; this is the same bound obtained by Trainor on BRK-type sets of degree ℓ, and we obtain this bound independently of 𝑑. 2𝑛 𝑛in the specific case of conical Kakeya sets: sets containing many hyperbolas or parabolas.