NOTE ON THE PINNED DISTANCE PROBLEM OVER FINITE FIELDS

Let 𝔽_q be a finite field with odd q elements. In this article, we prove that if E ⊆ 𝔽^d_q, d ≥ 2, and |E| ≥ q, then there exists a set Y ⊆ 𝔽^d_q with |Y| ~ q^d such that for all y ∈ Y, the number of distances between the point y and the set E is ~ q. As a corollary, we obtain that for each set E ⊆ 𝔽^d_q with |E| ≥ q, there exists a set Y ⊆ 𝔽^d_q with |Y| ~ q^d so that any set E ∪ {y} with y ∈ Y determines a positive proportion of all possible distances. The averaging argument and the pigeonhole principle play a crucial role in proving our results.