Note on the pinned distance problem over finite fields

Note on the pinned distance problem over finite fields

Let Fq be a finite field with odd q elements. In this article, we prove that if E ⊆ Fqd , d ≥ 2, and |E| ≥ q, then thered with |Y | ∼ qd such that for all y ∈ Y , theqexists a set Y ⊆ Fnumber of distances between the point y and the set E is ∼ q. Asd with |E| ≥ q, thereqa corollary, we obtain that for each set E ⊆ Fd with |Y | ∼ qd so that any set E ∪ {y} withqexists a set Y ⊆ Fy ∈ Y determines a positive proportion of all possible distances. The averaging argument and the pigeonhole principle play a crucial role in proving our results.